Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.2 Part 1 Right Triangle Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Use right triangles to evaluate trigonometric functions. Find function values for Recognize and use fundamental identities. Use equal cofunctions of complements. Evaluate trigonometric functions with a calculator. Use right triangle trigonometry to solve applied problems.

Labeling the Sides in Right Triangles : The segment across from the right angle ( ) is labeled the hypotenuse “Hyp.”. The “angle of interest” determines how to label the sides. Segment opposite from the Angle of interest( ) is labeled “Opp.” Segment adjacent to (next to) the Angle of interest ( ) is labeled “Adj.”. ** The angle of Interest is never the right angle.** Hyp. Opp. Angle of Interest (given or calculating) Adj.

Development of Right Triangle Definitions of Trigonometric Functions Let ABC represent a right triangle with right angle at C and angles A and B as acute angles, with side “a” opposite A, side “b” opposite B and side “c” (hypotenuse) opposite C. Place this triangle with either of the acute angles in standard position (in this example “A”): Notice that (b,a) is a point on the terminal side of A at a distance “c” from the origin

The Six Trigonometric Functions The six trigonometric functions are: Function Abbreviation sine sin cosine cos tangent tan cosecant csc secant sec cotangent cot

Right Triangle Definitions of Trigonometric Functions Based on this diagram, each of the six trigonometric functions for angle A would be defined: csc is the reciprocal of sin sec is the reciprocal of cos cot is the reciprocal of tan

Right Triangle Definitions of Trigonometric Functions The same ratios could have been obtained without placing an acute angle in standard position by making the following definitions: Standard “Right Triangle Definitions” of Trigonometric Functions ( MEMORIZE THESE!!!!!! )

Right Triangle Definitions of Trigonometric Functions S OH C AH T OA Sine Cosine Tangent 𝑎𝑑𝑗 ℎ𝑦𝑝 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑜𝑝𝑝 𝑎𝑑𝑗

Right Triangle Definitions of Trigonometric Functions In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle.

Example: Finding Trig Functions of Acute Angles Find the values of sin A, cos A, and tan A in the right triangle shown. A C B 52 48 20

Example: Evaluating Trigonometric Functions Find the value of the six trigonometric functions in the figure. We begin by finding c.

Calculate the 6 trigonometric functions for  . Your Turn: Calculate the 6 trigonometric functions for  . 20  48 First calculate the length of the hypotenuse sin  = csc  = cos  = sec  = tan  = tan  =

Side Lengths of Special Right ∆s Right triangles whose angle measures are 45° - 45° - 90° or 30° - 60° - 90° are called special right triangles. The theorems that describe the relationships between the side lengths of each of these special right triangles are as follows:

45˚-45˚-90˚ Triangle 45°-45°-90° Triangle In a 45°-45°-90° triangle, the legs s are congruent and the length of the hypotenuse ℎ is 2 times the length of a leg. s

30˚-60˚-90˚ Triangle 30°-60°-90° Triangle In a 30°-60°-90° triangle, the length of the hypotenuse ℎ is 2 times the length of the shorter leg s, and the length of the longer leg is √3 times the length of the shorter leg. s 2s s√3

Function Values for Some Special Angles A right triangle with a 45°, or 𝜋 4 radian, angle is isosceles – that is, it has two sides of equal length. The way in which we will use it is the hypotenuse will be the radius of the unit circle or 1 unit in length and the other two sides will be 1 2 or 2 2 . 1 unit 2 2 2 2

Function Values for Some Special Angles (continued) A right triangle that has a 30°, or 𝜋 6 radian, angle also has a 60°, or 𝜋 3 radian angle. In a 30-60-90 triangle, the measure of the side opposite the 30° angle is one-half the measure of the hypotenuse. The way in which we will use it is the hypotenuse will be the radius of the unit circle or 1 unit in length and the others sides will be ½ and 3 2 . 1 unit 3 2 ½

Example: Evaluating Trigonometric Functions of 45° Use the figure to find csc 45°, sec 45°, and cot 45°.

Calculate the 6 trigonometric functions for a 45 angle. Your Turn: Calculate the 6 trigonometric functions for a 45 angle. 1 45

Example: Evaluating Trigonometric Functions of 30° and 60° Use the figure to find tan 60° and tan 30°. If a radical appears in a denominator, rationalize the denominator. = 1 3 = 1 3 ∙ 3 3 = 3 3

Your Turn 1 Calculate the 6 trigonometric functions for a 60 angle. 60○ Calculate the 6 trigonometric functions for a 60 angle.

Trigonometric Functions of Special Angles

Usefulness of Knowing Trigonometric Functions of Special Anlges: 30o, 45o, 60o The trigonometric function values derived from knowing the side ratios of the 30-60-90 and 45-45-90 triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator. You will often be asked to find exact trig function values for angles other than 30o, 45o and 60o angles that are somehow related to trig function values of these angles.