Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 4.3 Right Triangle Trigonometry 1 Objective By following instructions students will be able to: 2 Evaluate trigonometric functions of acute angles. Use the fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real life problems. Recall: Soh-Cah-Toa 3 The Six Trigonometric Functions 4 Let be an acute angle of a right triangle. Then the six trigonometric functions of the angle are defined as follows. opp sin hyp adj cos hyp hyp csc opp opp tan adj hyp sec adj adj cot opp Example 1: Evaluating Trigonometric Functions Find the exact values of the six trigonometric functions of as shown below. hyp 3 4 5 Example 2: Evaluating Trigonometric Functions Find the exact values of sin 45, cos45 , and tan 45. 6 Example 3: Evaluating Trigonometric Functions of 30 and 60 degrees. Use the equilateral triangle shown to find the exact values of , , and . cos60 sin 30 cos30 7 sin 60 8 U-Try#1 1) Sketch a right triangle corresponding to the trig function of the acute angle theta. Use the Pythagorean Theorem to determine the length of the third side. Find the other trig functions of theta. a) 3 cos 7 b) cot 5 1 3 sin 30 tan 30 2) Use the given function to find: 2 and 3 b) cos30 a) csc 30 c) cot 60 d) cot 30 Sines, Cosines, and Tangents of Special Angles 9 1 2 3 cos30 cos 6 2 3 tan 30 tan 6 3 2 sin 45 sin 4 2 2 cos 45 cos 4 2 tan 45 tan 1 4 sin 30 sin 6 3 cos60 cos 1 tan 60 tan 3 sin 60 sin 3 3 2 3 2 Cofunctions of Complementary Angles are =. sin( 90 ) cos cos(90 ) sin tan(90 ) cot cot(90 ) tan sec(90 ) csc csc(90 ) sec 10 11 Fundamental Trigonometric Identities Reciprocal Identities 1 sin csc 1 cos sec 1 tan cot 1 csc sin 1 sec cos 1 cot tan Fundamental Trigonometric Identities 12 Quotient Identities sin tan cos cos cot sin Pythagorean Identities sin 2 cos2 1 1 tan 2 sec 2 1 cot 2 csc 2 13 Example 4: Applying Trigonometric Identities sin 0.6 Let be an acute angle such that . Find the values of (a) and (b) using trigonometric identities. tan cos Candy Crush 14 1. Look for clues and patterns (4 or 5 candies that are the same). 2. More than one way to complete the level 3. One objective: clear the jellies. 15 Candy Crush No more lives? • • • • • Get frustrated. Wait Look online for help. Ask friends for help. attempt again. Trigonometric Proofs 16 1. Look for clues and patterns (trig identities). 2. More than one way to complete the problem. 3. One objective: LHS=RHS 17 Trigonometric Proofs No more lives? • • • • • Get frustrated. Wait Look online for help. Ask friends for help. attempt again. Example 5: Using Trigonometric Identities 18 Use trigonometric identities to transform one side of the equation into the other. a) cos sec 1 b) (sec tan )(sec tan ) 1 Example 6: Using a Calculator Use a calculator to evaluate sec(540'12'') . 19 Example 7: Solving a Right Triangle A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5 degrees. How tall is the tree? 20 Example 8: Solving a Right Triangle A person is 200 yards from a river. Rather than walk directly to the river, the person walks 400 yards along a straight path to the river’s edge. Find the acute angle between this path and the river’s edge. 21 Example 9: Solving a Right Triangle A skateboard ramp requires a rise of one foot for each three feet of horizontal length. Find the lengths of sides b and c and find the measure of . 22 Revisit Objective Did we… 23 Evaluate trigonometric functions of acute angles? Use the fundamental trigonometric identities? Use a calculator to evaluate trigonometric functions? Use trigonometric functions to model and solve real life problems? Homework 24 Pg 310 #s 1-67 ALL