Acute Triangle Trigonometry

Math 20 Foundations Chapter 3: Acute Triangle Trigonometry &

Chapter 4: Oblique Triangle Trigonometry Outcome FM20.5  Demonstrate understanding of the cosine law and sine law (including the ambiguous case).

Indicators: a. Identify and describe situations relevant to self, family, or community that involve triangles without a right angle.  4.1, 3.2 b. Develop, generalize, explain, and apply strategies for determining angles or side lengths of triangles without a right angle.  3.1, 4.1 c. Draw diagrams to represent situations in which the cosine law or sine law could be used to solve a question.  3.2, 3.3, 3.4, 4.2 d. Explain the steps in a given proof of the sine law or cosine law.  3.1, 3.2, 3.3 e. Illustrate and explain how one, two, or no triangles could be possible for a given set of measurements for two side lengths and the non-included angle in a proposed triangle.  4.3 f. Develop, generalize, explain, and apply strategies for determining the number of solutions possible to a situation involving the ambiguous case.  4.3 g. Solve situational questions involving triangles without a right angle.  3.2, 3.3, 3.4 Chapter 3 & 4 Definitions

______: In any acute triangle,

______: In any acute triangle, a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C

______: A triangle that does not contain a 90° angle.

______case of the sine law: A situation in which two triangles can be drawn, given the available information; the ambiguous case may occur when the given measurements are the lengths of two sides and the measure of an angle that is not contained by the two sides (SSA). Trigonometry Review

Draw a right triangle and label the following parts:  Right angle  Short leg  Long leg  Hypotenuse

Label the triangle above , label the vertices using capital letters and label the sides using lower case letters. The right angle should be called , and the hypotenuse should be called .

 When you know the lengths of two sides of a right triangle, you can find the length of the third side using the Pythagorean Theorem:

where ‘’ is the length of the hypotenuse

Example 1 – Find the lengths of the indicated sides.

Review Trig Functions: SOH (sin=opp/hyp) CAH (cos=adj/hyp) TOA(tan=opp/adj)

Label the opposite side, adjacent side and hypotenuse with respect to angle .

Example 2- Determine the value of each trigonometric ratio to four decimal places. a) sin 33o b) cos 27o

Example 3- Determine the measure of A to the nearest degree. a) sin A = 0.8660 b) cos A = 0.8660 Example 4- Determine the value of x in each proportion. a) b)

Calculating an unknown side given an angle and one known side: Example 5- Find the value of .

Calculating an unknown angle given two known sides: Example 6- Find the value of .

To “solve” a triangle means to find all of the unknown sides and angles. Example 7- Solve the following triangle. Chapter 3: Acute Triangle Trigonometry Section 3.1: Exploring Side–Angle Relationships in Acute Triangles You have used the primary trigonometric ratios to determine side lengths and angle measures in right triangles.

Recall: SOH CAH TOA  In the figure to the right the ______are represented by lower case letters and the ______are represented by upper case letters  The hypotenuse is the side ______from the right angle in a triangle. Opposite means the side ______from the angle we are using. Adjacent means the side ______the angle we are using (but not the hypotenuse). SOH:

Can you use primary trigonometric ratios to determine unknown sides and angles in all acute triangles? The following diagram represents a general acute triangle.

A. What are two equivalent expressions that represent the height of ? B. If you drew the height of from a different vertex, how would the expressions for that height be different? 

C. Create an equation using the expression you created in part A. Show how your equation can be written as a ratio so that each ratio in the equation involves a side and an angle.  to write this as a ratio, we can divide both sides by sin B

D. Show how you could determine the measure of in this acute triangle.

**Make sure your calculator is set on DEG (degrees)**

Key Ideas:  The ratios of are ______for all three side–angle pairs in an acute triangle.  In an acute triangle, ∆ ABC,

3.1 Assignment: Nelson Foundations of Mathematics 11, Sec 3.1, pg. 117 Questions: 2, 4, workbook example 3a Section 3.2: Proving and Applying the Sine Law In Lesson 3.1, you discovered a side–angle relationship in acute triangles. Before this relationship can be used to solve problems, it must be proven to work in all acute triangles. Consider Ben’s proof: Example 1 – A triangle has angles measuring 80o and 55o. The side opposite the 80o angle is 12.0 m in length. Determine the length of the side opposite the 55o angle to the nearest tenth of a meter.

Example 2 – Toby uses chains attached to hooks on the ceiling and a winch to lift engines at his father’s garage. The chains, the winch, and the ceiling are arranged as shown. Toby solved the triangle using the sine law to determine the angle that each chain makes with the ceiling to the nearest degree. He claims that and . Is he correct? Explain, and make any necessary corrections. Key Ideas:  The sine law can be used to determine unknown side lengths or angle measures in acute triangles.  You can use the sine law to solve a problem modeled by an acute triangle when you know: o ______sides and the angle ______a known side. o two ______and any ______.  If you know the measures of two angles in a triangle, you can determine the third angle because the angles must add to ______.  When determining ______, it is more convenient to use:  When determining ______, it is more convenient to use:

3.2 Assignment: Nelson Foundations of Mathematics 11, Sec 3.2, pg. 124-127  Questions: 3abc, 4, 5, 6ac, 8a, 15 Section 3.3 Proving and Applying the Cosine Law

The sine law cannot always help you determine unknown angle measures or side lengths. Consider these triangles:

There are two unknowns in each pair of equivalent ratios, so the pairs cannot be used to solve for the unknowns. Another relationship is needed. This relationship is called the ______law, and it is derived from the Pythagorean Theorem. Before this relationship can be used to solve problems, it must be proven to work in all acute triangles: 1. Start by drawing an acute triangle ∆ABC. Draw the height from A to BC and label the intersection D. We will label BD=x and CD = y. o Why did we draw the height?

2. Write two different expressions for height (or h2) using the Pythagorean Theorem.

4. Set the expressions to equal one another and solve for c2

o Why can we set the two expressions to equal one another?

5. Rewrite the expression using only variables a, b, c and y and simplify:

 Why did we eliminate the variable x?

6. We must replace y. To do this we need to create an equivalent expression for y using only variables from ∆ABC (try using a trig function):

7. Substitute the expression you just developed in your previous equation:

Cosine Law: c2 = a2 + b2 - 2ab cos C

This form of the cosine law is useful because when you are given information about an acute triangle, you usually get information about the length of its sides or the size of its angles. This form we have developed includes only variables related to the sides and one angle of the triangle.

Let’s take another look at the triangles we examined at the beginning of the lesson. We were not able to use the sine law to determine the missing parts of the triangles, but now that we have learned the cosine law we may be able solve them.  Show how you can use the cosine law to determine the unknown side in .

 Show how you can use the cosine law to determine the unknown in .

Example 1 – Determine the length of to the nearest metre.

Example 2 – The diagram at the right shows the plan for a roof, with support beam parallel to . The local building code requires the angle formed at the peak of a roof to fall within a range of 70o to 80o so that snow and ice will not build up. Will this plan pass the local building code? First we need to make sure that the units are consistent (all feet)

Key Ideas  The Cosine law can be used to determine an unknown side length or angle measure in an acute triangle. a2 = b2 = c2 =

 You can use the cosine law to solve a problem that can be modeled by an acute triangle when you know: two sides and the all three sides. contained angle

o The contained angle is the angle ______two known sides.  When using the cosine law to determine an angle, you can: o substitute the known values first, then solve for the unknown angle. o rearrange the formula to solve for the cosine of the unknown angle, then substitute and evaluate.

3.3 Assignment: Nelson Foundations of Mathematics 11, Sec 3.3, pg. 136-139  Questions: 1, 4, 5, 6ac, 7ac, 8

Section 3.4: Solving Problems Using Acute Triangles

Example 1 – Two security cameras in a museum must be adjusted to monitor a new display of fossils. The cameras are mounted 6 m above the floor, directly across from each other on opposite walls. The walls are 12 m apart. The fossils are displayed in cases made of wood and glass. The top of the display is 1.5 m above the floor. The distance from the camera on the left to the centre of the top of the display is 4.8 m. Both cameras must aim at the centre of the top of the display. Determine the angles of ______, to the nearest degree, for each camera. 1. Draw a diagram. Be careful about where you put the display case. Label the angles of depression using Ѳ and α.

2. Use a trig function and the info you have to determine the size of Ѳ.

3. Use one of the laws we have learned about to determine the length of BD (or a).

4. You should now be able to use a trig function and the info we have for the right triangle on the right to determine the size of α. Example 2 – The world’s tallest free-standing totem pole is located in Beacon Hill Park in Victoria, British Columbia. While visiting the park, Manuel wanted to determine the height of the totem pole, so he drew a sketch and made some measurements:

(i) I walked along the shadow of the totem pole and counted 42 paces, estimating that each pace was about 1 m. (ii) I estimated that the angle of elevation of the Sun was about 40o. (iii) I observed that the shadow ran uphill, and I estimated that the angle the hill made with the horizontal was about 5o. Example 3 – Brendan and Diana plan to climb the cliff at Dry Island Buffalo Jump, Alberta. They need to know the height of the climb before they start. Brendan stands at point B, as shown in the diagram. He determines that , the angle of elevation to the top of the cliff, is 76o. Then he estimates , the angle between the base of the cliff, himself, and Diana, who is standing at point D. Diana estimates , the angle between the base of the cliff, herself, and Brendan. Determine the height of the cliff to the nearest metre.

Key Ideas:  The sine law, the cosine law, the primary trigonometric ratios, and the sum of angles in a triangle may all be useful when solving problems that can be modeled using acute triangles.  Drawing a clearly labeled diagram makes it easier to select a strategy for solving a problem.  To decide whether you need to use the sine law or the cosine law, consider the information given about the triangle and the measurement to be determined.

3.4 Assignment: Nelson Foundations of Mathematics 11, Sec 3.4, pg. 147-150 Questions: 3, 4, 5, 7, 9, 13

Chapter 4: Oblique Triangle Trigonometry Section 4.1: Exploring the Primary Trigonometric Ratios of Obtuse Angles  Until now, w have only worked with acute angles. We have used the trig ratios to determine the side lengths and angle measures in ______triangles, and we have used the sine and cosine laws to determine the side lengths and angle measures in acute ______triangles. Use a calculator to complete the table: sin (180o – cos (180o – tan (180o – sin cos tan ) ) )

What relationships do you observe when comparing the trig ratios for obtuse angles with the trig ratios for the related supplementary acute angles?  The sine ratios for supplementary angles are equal.  The cosine and tangent ratios for supplementary angles are opposites. Key Ideas  There are relationships between the value of a primary trigonometric ratio for an acute angle and the value of the same primary trigonometric ratio for the supplement of the acute angle.  For any acute angle, , sin = cos = tan = 4.1 Assignment: Nelson Foundations of Mathematics 11, Sec 4.1, pg. 163: Questions: 1-4 Section 4.2: Proving and Applying the Sine and Cosine Laws for Obtuse Triangles We have already shown that the sine law works for acute triangles. Now we are going to try to prove the sine law for obtuse triangles. Follow the steps of the investigation to prove that the sine law also applies to obtuse triangles. A. Draw an obtuse triangle ABC with height AD.

B. Write equations for sin (180o – ) and sin C using the two right triangles.

C. Use the transitive property to make the two expressions for AD equal to each other, then create a ratio.

D. Draw a new height, h, from B to base b in the triangle.

E. Write equations for sin A and sin C using the two right triangles.

F. Use the transitive property to make the two expressions for h equal to each other.

Example 1 – In an obtuse triangle, B measures 23.0o and its opposite side, b, has a length of 40.0 cm. Side a is the longest side of the triangle, with a length of 65.0 cm. Determine the measure of to the nearest tenth of a degree. In lesson 3.3 we proved the cosine law for acute triangles. Follow the steps of the investigation to prove that the cosine law also applies to obtuse triangles.

A. Use as shown below.

B. Extend the base of the triangle to D, creating two overlapping right triangles, and , with height BD. Note on your diagram that two angles are formed at C, and , such that 180o – . C. Let side CD be x. Use the Pythagorean Theorem to write two expressions for h2, using the two right triangles.

D. Use the transitive property to make the two expressions for h2 equal to each other. Re- arrange to isolate c2.

E. In the small right triangle, use a primary trig ratio to write an expression for x.

F. Substitute your expression for x into your equation from part D.

Remember that cos (180° - ACB) = -cos ACB:

Example 3 – The roof of a house consists of two slanted sections as shown. A roofing cap is being made to fit the crown of the roof, where the two slanted sections meet. Determine the measure of the angle needed for the roofing cap, to the nearest tenth of a degree. Key Ideas:  The sine law and cosine law can be used to determine unknown side lengths and angle measures in obtuse triangles.  The sine law and cosine law are used with obtuse triangles in the same way that they are used with acute triangles.  Be careful when using the sine law to determine the measure of an angle. The inverse sine of a ratio always gives an acute angle, but the supplementary angle has the same ratio. You must decide whether the acute angle, , or the obtuse angle 180o – is the correct angle for your triangle.  The measures of the angles determined using the cosine law are always correct. 4.2 Assignment: Nelson Foundations of Mathematics 11, Sec 4.2, pg. 170  Questions: 1, 2abc, 3ab, 4ab, 6, 9, 12 Section 4.3: The Ambiguous Case of the Sine Law SSA is a case where you are given the values for two sides of a triangle and the angle opposite one of the given sides:

The number of possible solutions to problems like this depends on the length of the sides given and the height of the triangle.  If the opposite side to the ______angle is ______than the height, then there is ______possible solution.  If the opposite side is ______than the height but still ______than the other given side, then there are _____ possible triangles.  If the opposite side is ______than the height and ______than the other given side, then there is ______possible triangle.  If the opposite side is the ______length as the height there is _____ possible solution. Example 1 – Given each SSA situation for , determine how many triangles are possible.

(a) = 30o, a = 4 m, b = 12 m (b) = 30o, a = 6 m, b = 12 m (c) = 30o, a = 8 m, b = 12 m (d) = 30o, a = 15 m, b = 12 m Example 2 – in obtuse ∆ABC, B = 24o, b = 18cm and a = 22cm. calculate the measure of A to the nearest degree. Is there more than one possible answer?

Example 3 – Martina and Carl are part of a team that is studying weather patterns. The team is about to launch a weather balloon to collect data. Martina’s rope is 7.8 m long and makes an angle of 36.0o with the ground. Carl’s rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical plane with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of a metre? Step 1 – Draw the triangle. Is it an SSA situation? If so, find the height of the triangle. Step 2 – Determine the number of possible triangles. Draw a sketch of all possible triangles. Step 3 – Solve the problem for ALL situations in Step 2. Situation 1: Situation 2: Remember: sin = sin (180o – Ѳ) Star by finding C for each situation then solving for x

Now solve for B to determine x

Key Idea  The ______case of the sine law may occur when you are given two side lengths and the measure of an angle that is ______one of these sides (SSA). Depending on the measure of the given angle and the lengths of the given sides, you may need to construct and solve ______, ______or ______triangles.  In , when h is the height of the triangle, and and the lengths of sides a and b are given, and is ______, there are four possibilities to consider:  In , when h is the height of the triangle, and and the lengths of sides a and b are given, and is ______, there are two possibilities to consider: 4.3 Assignment:pg. 183-185, Questions: 1ac, 2ace, 3, 4ad, 5, 6, 8