U6L1: Right Triangles and Trig Functions (Book 4.3) 1.23.17

I. The Six Trigonometric Functions

After spending many years staring at right triangles, ancient mathematicians discovered that if they made triangles with the same sized angles, the ratios between the sides of those triangles were always the same. Eventually they named those ratios, giving us the trig functions that you see below.

Answer these questions!

1) What is the relationship between the functions in the first row and the functions in the second row?

2) You can use the acronym “SOHCAHTOA” to remember these definitions. Write what each letter stands for.

3) Use the triangle above to find and . How are these ratios the same? How are they different?

4) Use the triangle above to find and. How are these ratios the same? How are they different?

II. Using Trig Functions to find the missing sides of right triangles Understanding these relationships formed by the six basic trig functions allows us to find missing parts of a right triangle. Unfortunately, these relationships only make sense with numbers that make sense as part of a right triangle. For example, it doesn’t make sense if the angle is 197° or -23°. So the ancient Greeks could only apply these ideas to triangles.

A. Special Right Triangles

For most right triangles, the ratios between the sides are awkward numbers – difficult to remember and difficult to work with. These two triangles are considered “special” because the ratios between their sides are easy to remember and calculate. They also include commonly used angles (45o is exactly half of a right angle, 30o is 1/3 of a right angle, 60o is 2/3 of a right angle.)

Answer these questions!

1) Calculate sine, cosine, and tangent for 300. What happens to the “x’s”?

2) Calculate sin45o and cos45o. Why are these the same?

3) Calculate tan60o and cot60o. Remember to rationalize denominators when possible.

B. Not Special Right Triangles

Use a calculator to help you find the values of trig functions for any angle that is not “special”. (NOTE: If the angle is measured in degrees, make sure your calculator is in degree mode.)

Ex: Find the value of x. III. “Undoing” Trig Functions to Find Angles

It is possible to “undo” trig functions using the concept of “inverse functions” or “working backwards”.

Ex1: Find the measurement of angle A, if angle A is an acute angle in a right triangle and .

How this is written How you find the answer

Ex 2: Find the measurement of angle A, if angle A is an acute angle in a right triangle and .

How this is written How you find the answer

IV. Problems that can be solved using right triangle trig.

A) Find the missing side

Ex: A surveyor measures the angle from the ground to the top of a mountain from a distance of 3 miles. Find the height of the mountain if the angle was 23o.

Why would anyone use this method?

B) Find the angle

Ex: A motion detector is placed on the roof of a 50 ft building in such a way that an alarm will sound if anything crosses through its line of sight. What angle must be formed between the detector and the side of the building so that the alarm will go off at a distance of 20 feet?

Why would anyone use this method?

V. Checking Your Answers I will provide you with a key whenever I can, but sometimes you will need to be able to check your own work. Fortunately, there are a few simple, common sense things you can do to check your work.

A) Use facts about right triangles

If your answer breaks any of these rules, then you KNOW it is wrong.

1) The two acute angles in a right triangle must always add up to ______.

2) The longest side of a right triangle is always the ______.

3) The two shorter sides of a right triangle must always add up to more than the longest side.

4) Shorter sides are always across from smaller angles.

B) Word Problems & Common Sense

1) There is no guarantee that every word problem will be based on realistic numbers, but if you get an answer that seems surprisingly big or small, double check your math.

2) Very often there are realistic limits for word problems. For example, a distance will never be a negative number. Likewise, when calculating compound interest the amount should always grow. Think about limits like these and double check your math if they don’t match your answer. WHAT HAVE YOU DONE !?!? Results from U6L1 Practice (Odds)

Find someone that you did NOT talk with while you worked on your practice questions and answer the following discussion questions:

1) Were there any questions that you did differently? If so, explain how the work was different and if it was correct. If you did everything the same way, write an explanation of what you did for one question of your choice.

2) Did either of you get any questions wrong? What are some things you should consider if you want to avoid mistakes on these questions on a quiz? (Be specific.)

3) Today’s goal was “To solve application problems involving triangles”. Do you think you have met that goal? Is there anything else that either of you think you need to know before you finish the even numbered questions at home?