Lesson 7.2 Notes Complete PDF

Chapter 7: The Unit Circle: Sine and Cosine Functions Page |8 Learning Objectives: Section 7.2 Right Triangle Trigonometry In this section, you will: Use right triangles to evaluate trigonometric functions...

Chapter 7: The Unit Circle: Sine and Cosine Functions Page |8 Learning Objectives: Section 7.2 Right Triangle Trigonometry In this section, you will: Use right triangles to evaluate trigonometric functions 𝜋 𝜋 𝜋 Find functions values for 30∘ ( ) , 45∘ ( ) , 60∘ ( ). 6 4 3 Use equal cofunctions of complementary angles. Use the definitions of trigonometric functions of any angle. Use right-triangle trigonometry to solve applied problems. Using Right Triangles to Evaluate Trigonometric Functions A triangle inscribed in the unit circle Right Acute triangle ((0)(t) sin(t) , Given a right triangle with an acute angle of t, the first three trigonometric functions are listed. opposite adjacent opposite Sine sin 𝑡 = Cosine cos 𝑡 = Tangent tan 𝑡 = hypotenuse hypotenuse adjacent 𝑜 𝑎 𝑜 𝑆 𝐶 𝑇 ℎ ℎ 𝑎 Using the information from the first triangle we can define sine, cosine and tangent using the unit circle. sin 𝑡 = 4 = 3 cos 𝑡 = * = x tan 𝑡 = = Example Evaluating a Trigonometric Function of a Right Triangle a. Given the triangle shown in Figure 3, find the value of cos 𝛼. cos = b. Given the triangle shown in Figure 4, find the value of sin 𝑡. sint t =  Chapter 7: The Unit Circle: Sine and Cosine Functions Page |9 Reciprocal Functions hypotenuse hypotenuse adjacent Secant sec 𝑡 = Cosecant csc 𝑡 = Cotangent cot 𝑡 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 opposite opposite * & * * Example 2 Evaluating Trigonometric Functions of Angles Not in Standard Position a. Using the triangle shown in Figure 6, evaluate sin 𝛼 , cos 𝛼 , tan 𝛼 , sec 𝛼 , csc 𝛼 , and cot 𝛼 sin = (xc = - sec = 1 = a coa = tunc = 1 co + c = 3 b. Using the triangle shown in Figure 7, evaluate sin 𝑡 , cos 𝑡 , tan 𝑡 , sec 𝑡 , csc 𝑡 , and cot 𝑡 S · cot + = E Finding Trigonometric Functions of Special Angles Using Side Lengths TY4 T/6 af a as 2a 7 TY4 7 T a a Example 3 Evaluating Trigonometric Functions of Special Angles Using Side Lengths 𝜋 a. Find the exact value of the trigonometric functions of 3 , using side lengths. sin Cs = cos t see = 2 tant = Es cut #= 𝜋 b. Find the exact value of the trigonometric functions of 4 , using side lengths. sir s = E cos see tan = cut F = 1  Chapter 7: The Unit Circle: Sine and Cosine Functions P a g e | 10 Using Equal Cofunctions of Complements Example 4 Using Cofunctions Identities a. If sin 𝑡 = 12, find cos ( 2 − 𝑡).= 5 𝜋 𝜋 𝜋 b. If csc ( 6 ) = 2, find sec ( 3 ). = 2 Example 5 Finding Missing Side Lengths Using Trigonometric Ratios a. Find the unknown sides of the triangle in Figure 11. 14 753 Gjo 𝜋 b. A right triangle has one angle of and a hypotenuse of 20. Find the unknown sides and angle of the triangle. 3 loss 7 o Example 6 Measuring a Distance Indirectly a. To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of 57∘ between a line of sight to the top of the tree and the ground, as shown in Figure 13. Find the height of the tree. tanst =0 h= 30th 57 h b. How long of a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building 5𝜋 making an angle of 12 with the ground? Round to the nearest foot. sin so =