Right Triangle Trigonometry Unit 3 Day 2 Right Triangle Trigonometry

Trigonometric Functions of Acute Angles Determine the six trigonometric ratios for a given acute angle of a right triangle. Determine the trigonometric function values of 30º, 45º, and 60º. Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle. Given the function values of an acute angle, find the function values of its complement.

Right Triangles and Acute Angles Greek letters such as  (alpha),  (beta),  (gamma),  (theta), and  (phi) are often used to denote an angle. We label the sides with respect to angles. The hypotenuse is opposite the right angle. There is the side opposite  and the side adjacent to . opposite  adjacent to  hypotenuse 

Trigonometric Ratios The lengths of the sides of a right triangle are used to define the six trigonometric ratios: sine (sin) cosine (cos) tangent (tan) cosecant (csc) secant (sec) cotangent (cot) opposite  adjacent to  hypotenuse 

The six trigonometric functions of  are: …and their reciprocals (flip them over!)

Copyright © 2009 Pearson Education, Inc. Example   13 5 In the triangle shown, find the six trigonometric function values of (a)  and (b) . Solution: Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example   13 5 In the triangle shown, find the six trigonometric function values of (a)  and (b) . Solution: Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Reciprocal Functions Note that there is a reciprocal relationship between pairs of the trigonometric functions. Copyright © 2009 Pearson Education, Inc.

Example Given that find csc , sec , and cot . Solution:

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SOH CAH TOA

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Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the value of each of the six trigonometric functions of the angle . Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2009 Pearson Education, Inc. If and  is an acute angle, find the other five trigonometric function values of . Solution: Use the definition of the sine function that the ratio and draw a right triangle. Use the Pythagorean equation to find a. 7 a 6  Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Solution continued Use the lengths of the three sides to find the other five ratios. Copyright © 2009 Pearson Education, Inc.

Finding the Missing Pieces: If we are given two measurements in a right triangle we can find the other measurements of the triangle using trig ratios. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.) Solution: Draw the situation, label the acute angle and length of the adjacent side. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Solution continued: The balloon is approximately 0.7 mi, or 3696 ft, high. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Find the trigonometric function value, rounded to four decimal places, of each of the following: Solution: Check that the calculator is in degree mode. Copyright © 2009 Pearson Education, Inc.

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Copyright © 2009 Pearson Education, Inc. Example A paint crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position? Solution: Draw the situation, label the hypotenuse and length of the side adjacent to . Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Solution continued: Use a calculator to find the acute angle whose cosine is 0.26: Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example House framers can use trigonometric functions to determine the lengths of rafters for a house. They first choose the pitch of the roof, or the ratio of the rise over the run. Then using a triangle with that ratio, they calculate the length of the rafter needed for the house. Jose is constructing rafters for a roof with a 10/12 pitch on a house that is 42 ft wide. Find the length x of the rafter of the house to the nearest tenth of a foot. Rise: 10 Run: 12  Pitch: 10/12 Copyright © 2009 Pearson Education, Inc.

First find the angle  that the rafter makes with the side wall.  ≈ 39.8º Use the cosine function to determine the length x of the rafter.

Copyright © 2009 Pearson Education, Inc. Solution continued: x 39.8º 21 ft The length of the rafter for this house is approximately 27.3 ft. Copyright © 2009 Pearson Education, Inc.