DOCSLIB.ORG

How to Learn Trigonometry Intuitively | Betterexplained 9/26/15, 12:19 AM

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
How to Learn Trigonometry Intuitively | Betterexplained 9/26/15, 12:19 AM How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (/) How To Learn Trigonometry Intuitively by Kalid Azad · 101 comments Tweet 73 Trig mnemonics like SOH-CAH-TOA (http://mathworld.wolfram.com/SOHCAHTOA.html) focus on computations, not concepts: TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half- dedicated to vision processing, seem to enjoy imagery. And “TOA” evokes the stunning beauty of an abstract ratio. I think you deserve better, and here’s what made trig click for me. Visualize a dome, a wall, and a ceiling Trig functions are percentages to the three shapes http://betterexplained.com/articles/intuitive-trigonometry/ Page 1 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM Motivation: Trig Is Anatomy Imagine Bob The Alien visits Earth to study our species. Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”. After creating specific terms for anatomy, Bob might jot down typical body proportions (http://en.wikipedia.org/wiki/Body_proportions): The armspan (fingertip to fingertip) is approximately the height A head is 5 eye-widths wide Adults are 8 head-heights tall http://betterexplained.com/articles/intuitive-trigonometry/ Page 2 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (http://en.wikipedia.org/wiki/Vitruvian_Man) How is this helpful? Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate the owner’s height. And head size. And eye width. One fact is linked to a variety of conclusions. Even better, human biology explains human thinking. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations. Now the plot twist: you are Bob the alien, studying creatures in math-land! Generic words like “triangle” aren’t overly useful. But labeling sine, cosine, and hypotenuse helps us notice deeper connections. And scholars might study haversine, exsecant and gamsin (http://www.theonion.com/articles/nations-math-teachers- http://betterexplained.com/articles/intuitive-trigonometry/ Page 3 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM introduce-27-new-trig-functi,33804/), like biologists who find a link between your fibia and clavicle. And because triangles show up in circles… …and circles appear in cycles, our triangle terminology helps describe repeating patterns! Trig is the anatomy book for “math-made” objects. If we can find a metaphorical triangle, we’ll get an armada of conclusions for free. Sine/Cosine: The Dome Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth. Pretend you’re in the middle of your dome, about to hang up a movie screen. You point to some angle “x”, and that’s where the screen will hang. http://betterexplained.com/articles/intuitive-trigonometry/ Page 4 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM The angle you point at determines: sine(x) = sin(x) = height of the screen, hanging like a sign cosine(x) = cos(x) = distance to the screen along the ground [“cos” ~ how “close”] the hypotenuse, the distance to the top of the screen, is always the same Want the biggest screen possible? Point straight up. It’s at the center, on top of your head, but it’s big dagnabbit. Want the screen the furthest away? Sure. Point straight across, 0 degrees. The screen has “0 height” at this position, and it’s far away, like you asked. The height and distance move in opposite directions: bring the screen closer, and it gets taller. Tip: Trig Values Are Percentages Nobody ever told me in my years of schooling: sine and cosine are percentages. They vary from +100% to 0 to -100%, or max positive to nothing to max negative. Let’s say I paid $14 in tax. You have no idea if that’s expensive. But if I say I paid 95% in tax, you know I’m getting ripped off. http://betterexplained.com/articles/intuitive-trigonometry/ Page 5 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM An absolute height isn’t helpful, but if your sine value is .95, I know you’re almost at the top of your dome. Pretty soon you’ll hit the max, then start coming down again. How do we compute the percentage? Simple: divide the current value by the maximum possible (the radius of the dome, aka the hypotenuse). That’s why we’re told “Sine = Opposite / Hypotenuse”. It’s to get a percentage! A better wording is “Sine is your height, as a percentage of the hypotenuse”. (Sine becomes negative if your angle points “underground”. Cosine becomes negative when your angle points backwards.) Let’s simplify the calculation by assuming we’re on the unit circle (radius 1). Now we can skip the division by 1 and just say sine = height. Every circle is really the unit circle, scaled up or down to a different size. So work out the connections on the unit circle and apply the results to your particular scenario. Try it out: plug in an angle and see what percent of the height and width it reaches: Sine and Cosine R1 x = 30 30 R2 sine 50 % of height R3 cosine 86.60254037844 % of width The growth pattern of sine isn’t an even line. The first 45 degrees cover 70% of the height, and the final 10 degrees (from 80 to 90) only cover 2%. This should make sense: at 0 degrees, you’re moving nearly vertical, but as you get to the top of the dome, your height changes level off. Tangent/Secant: The Wall http://betterexplained.com/articles/intuitive-trigonometry/ Page 6 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM One day your neighbor puts up a wall right next to your dome. Ack, your view! Your resale value! But can we make the best of a bad situation? Sure. What if we hang our movie screen on the wall? You point at an angle (x) and figure out: tangent(x) = tan(x) = height of screen on the wall distance to screen: 1 (the screen is always the same distance along the ground, right?) secant(x) = sec(x) = the “ladder distance” to the screen We have some fancy new vocab terms. Imagine seeing the Vitruvian “TAN GENTleman” projected on the wall. You climb the ladder, making sure you can “SEE, CAN’T you?”. (Yeah, he’s naked… won’t forget the analogy now, will you?) Let’s notice a few things about tangent, the height of the screen. http://betterexplained.com/articles/intuitive-trigonometry/ Page 7 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM It starts at 0, and goes infinitely high. You can keep pointing higher and higher on the wall, to get an infinitely large screen! (That’ll cost ya.) Tangent is just a bigger version of sine! It’s never smaller, and while sine “tops off” as the dome curves in, tangent keeps growing. How about secant, the ladder distance? Secant starts at 1 (ladder on the floor to the wall) and grows from there Secant is always longer than tangent. The leaning ladder used to put up the screen must be longer than the screen itself, right? (At enormous sizes, when the ladder is nearly vertical, they’re close. But secant is always a smidge longer.) Remember, the values are percentages. If you’re pointing at a 50-degree angle, tan(50) = 1.19. Your screen is 19% larger than the distance to the wall (the radius of the dome). Tangent and Secant R1 x = 50 50 R2 tangent 119.17535925942 % of hyp. R3 secant 155.57238268604 % of hyp. (Plug in x=0 and check your intuition that tan(0) = 0, and sec(0) = 1.) Cotangent/Cosecant: The Ceiling Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. (What’s with this guy? Oh, the naked-man-on-my-wall incident…) Well, time to build a ramp to the ceiling, and have a little chit chat. You pick an angle to build and work out: http://betterexplained.com/articles/intuitive-trigonometry/ Page 8 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM cotangent(x) = cot(x) = how far the ceiling extends before we connect cosecant(x) = csc(x) = how long we walk on the ramp the vertical distance traversed is always 1 Tangent/secant describe the wall, and COtangent and COsecant describe the ceiling. Our intuitive facts are similar: If you pick an angle of 0, your ramp is flat (infinite) and never reachers the ceiling. Bummer. The shortest “ramp” is when you point 90-degrees straight up. The cotangent is 0 (we didn’t move along the ceiling) and the cosecant is 1 (the “ramp length” is at the minimum). Visualize The Connections A short time ago I had zero “intuitive conclusions” about the cosecant. But with the dome/wall/ceiling metaphor, here’s what we see: http://betterexplained.com/articles/intuitive-trigonometry/ Page 9 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM Whoa, it’s the same triangle, just scaled to reach the wall and ceiling.
Load more

Recommended publications
  • Elementary Functions Triangulation and the Law of Sines Three Bits of A
    Triangulation and the Law of Sines A triangle has three sides and three angles; their values provide six pieces of information about the triangle. In ordinary geometry, most of the time three pieces of information are Elementary Functions sufficient to give us the other three pieces of information. Part 5, Trigonometry Lecture 5.3a, The Laws of Sines and Triangulation In order to more easily discuss the angles and sides of a triangle, we will label the angles by capital letters (such as A; B; C) and label the sides by small letters (such as a; b; c.) Dr. Ken W. Smith We will assume that a side labeled with a small letter is the side opposite Sam Houston State University the angle with the same, but capitalized, letter. For example, the side a is opposite the angle A. 2013 We will also use these letters for the values or magnitudes of these sides and angles, writing a = 3 feet or A = 14◦: Smith (SHSU) Elementary Functions 2013 1 / 26 Smith (SHSU) Elementary Functions 2013 2 / 26 Three Bits of a Triangle AA and AAA Suppose we have three bits of information about a triangle. Can we recover all the information? AAA If the 3 bits of information are the magnitudes of the 3 angles, and if we If we know the values of two angles then since the angles sum to π have no information about lengths of sides, then the answer is No. (= 180◦), we really know all 3 angles. Given a triangle with angles A; B; C (in ordinary Euclidean geometry) we can always expand (or contract) the triangle to a similar triangle which So AAA is really no better than AA! has the same angles but whose sides have all been stretched (or shrunk) by some constant factor.
    [Show full text]
  • Mathematics Tables
    MATHEMATICS TABLES Department of Applied Mathematics Naval Postgraduate School TABLE OF CONTENTS Derivatives and Differentials.............................................. 1 Integrals of Elementary Forms............................................ 2 Integrals Involving au + b ................................................ 3 Integrals Involving u2 a2 ............................................... 4 Integrals Involving √u±2 a2, a> 0 ....................................... 4 Integrals Involving √a2 ± u2, a> 0 ....................................... 5 Integrals Involving Trigonometric− Functions .............................. 6 Integrals Involving Exponential Functions ................................ 7 Miscellaneous Integrals ................................................... 7 Wallis’ Formulas ................................................... ...... 8 Gamma Function ................................................... ..... 8 Laplace Transforms ................................................... 9 Probability and Statistics ............................................... 11 Discrete Probability Functions .......................................... 12 Standard Normal CDF and Table ....................................... 12 Continuous Probability Functions ....................................... 13 Fourier Series ................................................... ........ 13 Separation of Variables .................................................. 15 Bessel Functions ................................................... ..... 16 Legendre
    [Show full text]
  • Trigonometric Computation David Altizio, Andrew Kwon
    Trig Computation Western PA ARML 2016-2017 Page 1 Trigonometric Computation David Altizio, Andrew Kwon 0.1 Lecture • There are many different formulas for the area of a triangle. The ones that you'll probably need to know the most are summarized below: 1 1 abc A = bh = ab sin C = rs = = ps(s − a)(s − b)(s − c) = r (s − a): 2 2 4R a Make sure you know how to derive these! • Law of Sines: In any triangle ABC, we have a b c = = = 2R: sin A sin B sin C • Law of Cosines: In any triangle ABC, we have c2 = a2 + b2 − 2ab cos C: • Ratio Lemma: In any triangle ABC, if D is a point on BC, then BD BA sin BAD = \ : DC AC sin \DAC Note that this is a generalization of the Angle Bisector Theorem. Note further that this does not necessarily require that D lie on the segment BC. • Make sure you know your trig identities! Here are a few ones to keep in mind: { Negation: We have sin(−α) = − sin α and cos(−α) = cos α for all angles α. { Pythagorean Identity: For any angle α, sin2 α + cos2 α = 1: Be sure to memorize this one! { Sum and Difference Identities: For any angles α and β, sin(α + β) = sin α cos β + sin β cos α; cos(α + β) = cos α cos β − sin α sin β: { Double Angle Identities: We have 2 tan A sin(2A) = 2 sin A cos A; cos(2A) = cos2 A − sin2 A; tan(2A) = : 1 − tan2 A { Half Angle Identities: We have r r A 1 − cos A A 1 + cos A A r1 − cos A sin = ; cos = ; tan = : 2 2 2 2 2 1 + cos A Trig Computation Western PA ARML 2016-2017 Page 2 0.2 Problems 1.
    [Show full text]
  • SFTE FTE Reference Handbook
    SFTE Reference Handbook Third Edition 2013 Society of Flight Test Engineers Reference Handbook Third Edition 2013 Page - i SFTE Reference Handbook Third Edition 2013 Society of Flight Test Engineers Reference Handbook 2013 Edition Corporate support supplied by Cessna Aircraft for printing the 2007 Edition And The National Test Pilot School Contributing Authors Al Lawless (sections 1-8, 10-12, 15, 18) Greg Lewis (section 2.6) Bill Norton (sections 9, 13) Dan Hrehov (section 14) Steven Arney (section 16) John Minor (section 19) David Kidman, Christopher Moulder, Craig Stevens (section 17) Edited by Lee Gardner & Darcy Painter 1998-2006 Harold Weaver 2006-2013 The SFTE handbook committee continually seeks corporate sponsors for this book and authors for new sections (including but not limited to INS, GPS, EMI/EMF, radar, avionics, R&M, E-O, human factors, orbital mechan- ics, armament) Page - ii SFTE Reference Handbook Third Edition 2013 Publication Policy Copyright (C) 2013 by Society Of Flight Test Engineers All rights reserved. This Technical Handbook is for the exclusive use of the Society of Flight Test Engineers individual and Corporate Members. The Technical information contained herein may not be reproduced by any other individual or organization in any form without writ- ten permission from the Society of Flight Test Engineers. The Society reserves the exclusive right of publication. For further information concerning the publication policy, write to: Society of Flight Test Engineers 44814 N. Elm Avenue Lancaster, California 93534 USA Or: Contact the Society of Flight Test Engineers through their web site at www.sfte.org. Please submit corrections or additions to SFTE Handbook Committee 44814 N.
    [Show full text]
  • 13.4A Area Law of Sines
    Algebra 2 13.4A Area of Triangle, Law of Sines Obj: able to find exact values of trigonometric functions for general angles; able to use reference triangles Area of a Triangle B The area of any triangle is given by one half the product of the lengths of two sides times the sine of their included angle. 1 c Area = bc sin A a 2 1 Area = ab sin C 2 A 1 b C Area = ac sin B 2 Find the area of the triangle . 1. B 3 C 25 ° 6 A Oblique triangles are triangles that have no right angles . C Use Law of Sines to solve (AAS or ASA) a b Law of Sines If ABC is a triangle with sides a, b, and c, a b c sin A SinB SinC then = = or = = sin A sin B sin C a b c A c B 2. Given a triangle with A = °,70 B = 80 a , a = 30 in ., find the remaining side and angles. Round sides to the nearest tenth and angles to the nearest degree. There has to be an angle and a side across from each other to use the Law of Sines. So fill in the table below. A = a = B = b = C = c = 3. Given a triangle with B = 56 a ,C = 84 a ,b = 15 feet , find the remaining angle and sides. Round sides to the nearest tenth and angles to the nearest degree. There has to be an angle and a side across from each other to use the Law of Sines.
    [Show full text]
  • Chapter 8: Further Applications of Trigonometry in This Chapter, We Will Explore Additional Applications of Trigonometry
    Section 8.1 Non-Right Triangles: Laws of Sines and Cosines 451 Chapter 8: Further Applications of Trigonometry In this chapter, we will explore additional applications of trigonometry. We will begin with an extension of the right triangle trigonometry we explored in Chapter 5 to situations involving non-right triangles. We will explore the polar coordinate system and parametric equations as new ways of describing curves in the plane. In the process, we will introduce vectors and an alternative way of writing complex numbers, two important mathematical tools we use when analyzing and modeling the world around us. Section 8.1 Non-right Triangles: Law of Sines and Cosines ...................................... 451 Section 8.2 Polar Coordinates ..................................................................................... 467 Section 8.3 Polar Form of Complex Numbers ............................................................ 480 Section 8.4 Vectors ..................................................................................................... 491 Section 8.5 Parametric Equations ............................................................................... 504 Section 8.1 Non-Right Triangles: Laws of Sines and Cosines Although right triangles allow us to solve many applications, it is more common to find scenarios where the triangle we are interested in does not have a right angle. Two radar stations located 20 miles apart both detect a UFO located between them. The angle of elevation measured by the first station is 35 degrees. The angle of 15° 35° elevation measured by the second station is 20 miles 15 degrees. What is the altitude of the UFO? We see that the triangle formed by the UFO and the two stations is not a right triangle. Of course, in any triangle we could draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles, but it would be nice to have methods for working directly with non-right triangles.
    [Show full text]
  • Fractal Zoomer Help
    Fractal Zoomer Version: 1.0.6.7 Author: Christos Kalonakis Contact: [email protected] File Menu Starting Position, restores the default position and size for each fractal. Go To, lets you, accurate, choose the center and the size. When Julia option is selected for the first time, Go To lets you set the Julia's seed. Zoom In, lets you zoom in with a fixed zooming factor. Zoom Out, lets you zoom out with a fixed zooming factor. Up, lets you move one screen up. Down, lets you move one screen down. Left, lets you move one screen left. Right, lets you move one screen right. Save Settings As, saves the center, size, fractal color, palette, out coloring mode, in coloring mode, iterations, bailout, fractal function, plane, rotation, initial value, perturbation, julia settings, domain coloring, and image filters. Load Settings, loads the center, size, fractal color, palette, out coloring mode, in coloring mode, iterations, bailout, fractal function, plane, rotation, initial value, perturbation, julia settings, domain coloring, and image filters. Save Image As, lets you save a png image. Save Settings and Image As, lets you save the current settings and a png image. Edit User Code, opens the text editor and lets you edit the user code. Compile User Code, compiles the user code. Exit, exits the application. Options Menu Fractal Options Menu, lets you set the fractal options. Colors Menu, lets you set the color options. Iterations Menu, lets you change the maximum iterations. Bailout Conditions Menu, lets you set the bailout condition. Bailout, lets you set the value that will be checked through the bailout test algorithm.
    [Show full text]
  • Section 8.1 Non-Right Triangles: Laws of Sines and Cosines 451
    Section 8.1 Non-Right Triangles: Laws of Sines and Cosines 451 Chapter 8: Further Applications of Trigonometry In this chapter, we will explore additional applications of trigonometry. We will begin with an extension of the right triangle trigonometry we explored in Chapter 5 to situations involving non-right triangles. We will explore the polar coordinate system and parametric equations as new ways of describing curves in the plane. In the process, we will introduce vectors and an alternative way of writing complex numbers, two important mathematical tools we use when analyzing and modeling the world around us. Section 8.1 Non-right Triangles: Law of Sines and Cosines ...................................... 451 Section 8.2 Polar Coordinates ..................................................................................... 467 Section 8.3 Polar Form of Complex Numbers ............................................................ 480 Section 8.4 Vectors ..................................................................................................... 491 Section 8.5 Parametric Equations ............................................................................... 504 Section 8.1 Non-Right Triangles: Laws of Sines and Cosines Although right triangles allow us to solve many applications, it is more common to find scenarios where the triangle we are interested in does not have a right angle. Two radar stations located 20 miles apart both detect a UFO located between them. The angle of elevation measured by the first station is 35 degrees. The angle of 15° 35° elevation measured by the second station is 20 miles 15 degrees. What is the altitude of the UFO? We see that the triangle formed by the UFO and the two stations is not a right triangle. Of course, in any triangle we could draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles, but it would be nice to have methods for working directly with non-right triangles.
    [Show full text]
  • Law of Sines
    333202_0601.qxd 12/5/05 10:40 AM Page 430 430 Chapter 6 Additional Topics in Trigonometry 6.1 Law of Sines What you should learn •Use the Law of Sines to solve Introduction oblique triangles (AAS or ASA). In Chapter 4, you studied techniques for solving right triangles. In this section •Use the Law of Sines to solve and the next, you will solve oblique triangles—triangles that have no right oblique triangles (SSA). angles. As standard notation, the angles of a triangle are labeled A, B, and C, and •Find the areas of oblique their opposite sides are labeled a, b, and c, as shown in Figure 6.1. triangles. •Use the Law of Sines to model C and solve real-life problems. Why you should learn it b a You can use the Law of Sines to solve real-life problems involving A c B oblique triangles. For instance, in Exercise 44 on page 438, you FIGURE 6.1 can use the Law of Sines to determine the length of the To solve an oblique triangle, you need to know the measure of at least one shadow of the Leaning Tower side and any two other parts of the triangle—either two sides, two angles, or one of Pisa. angle and one side. This breaks down into the following four cases. 1. Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) The first two cases can be solved using the Law of Sines, whereas the last two cases require the Law of Cosines (see Section 6.2).
    [Show full text]
  • Precalculus: Law of Sines and Law of Cosines Concepts: Law of Sines
    Precalculus: Law of Sines and Law of Cosines Concepts: Law of Sines, Law of Cosines. Law of Sines The law of sines is used to determine all the angles and all the lengths of a general triangle given partial information about the triangle (which is in general not a right-angle triangle). H ¡ HH ¡ H ¡ h HH H ¡ HH ¡ H There are two possibilities for the shape of the triangle created with interior angles A; B; C and sides of length a; b; c. The sides are labelled opposite their corresponding angles. The perpendicular height is labelled h in both cases. C C HH ¡ H a HH ¡ b h H b ¡a h HH ¡ H A H A ¡ c B c B opp h From either of the diagrams above, we have sin A = = . hyp b opp h Also, from the diagram on the left, we have sin B = = . hyp a h Also, from the diagram on the right, we have sin(π − B) = . (remember, B is the interior angle!) a We need to show th sin B = sin(π − B) so we can use the Law of Sines for either situation. sin(u − v) = sin u cos v − cos u sin v start with trig identity sin(π − B) = sin π cos B − cos π sin B = (0) cos B − (−1) sin B h = sin B = a Therefore, for both situations (skinny triangle or fat triangle) we have sin A sin B h = b sin A = a sin B ) = a b You could do exactly the same thing where you drop the perpendicular to the other two sides.
    [Show full text]
  • Table of Contents - Unit 2 – Trigonometric Functions Table of Contents - Unit 2 – Trigonometric Functions
    TABLE OF CONTENTS - UNIT 2 – TRIGONOMETRIC FUNCTIONS TABLE OF CONTENTS - UNIT 2 – TRIGONOMETRIC FUNCTIONS ........................................................................................ 1 ESSENTIAL CONCEPTS OF TRIGONOMETRY ............................................................................................................................. 3 WHAT IS TRIGONOMETRY? .................................................................................................................................................................... 3 WHY TRIANGLES? ................................................................................................................................................................................. 3 EXAMPLES OF PROBLEMS THAT CAN BE SOLVED USING TRIGONOMETRY .............................................................................................. 3 GENERAL APPLICATIONS OF TRIGONOMETRIC FUNCTIONS .................................................................................................................... 3 EXTREMELY IMPORTANT PREREQUISITE KNOWLEDGE .......................................................................................................................... 3 TRIGONOMETRY OF RIGHT TRIANGLES – TRIGONOMETRIC RATIOS OF ACUTE ANGLES ........................................................................ 4 THE SPECIAL TRIANGLES – TRIGONOMETRIC RATIOS OF SPECIAL ANGLES ........................................................................................... 4 HOMEWORK ..........................................................................................................................................................................................
    [Show full text]
  • Section 7.3 - the Law of Sines and the Law of Cosines
    Section 7.3 - The Law of Sines and the Law of Cosines Sometimes you will need to solve a triangle that is not a right triangle. This type of triangle is called an oblique triangle. To solve an oblique triangle you will not be able to use right triangle trigonometry. Instead, you will use the Law of Sines and/or the Law of Cosines. You will typically be given three parts of the triangle and you will be asked to find the other three. The approach you will take to the problem will depend on the information that is given. If you are given SSS (the lengths of all three sides) or SAS (the lengths of two sides and the measure of the included angle), you will use the Law of Cosines to solve the triangle. If you are given SAA (the measures of two angles and one side) or SSA (the measures of two sides and the measure of an angle that is not the included angle), you will use the Law of Sines to solve the triangle. Recall from your geometry course that SSA does not necessarily determine a triangle. We will need to take special care when this is the given information. 1 Please read this! Here are some facts about solving triangles that may be helpful in this section: If you are given SSS, SAS or SAA, the information determines a unique triangle. If you are given SSA, the information given may determine 0, 1 or 2 triangles. This is called the “ambiguous case” of the law of sines.
    [Show full text]