Right Triangle Trigonometry

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1 Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use fundamental trigonometric identities, use a calculator to evaluate trigonometric functions, use trigonometric functions to model and solve real-world problems.

3 Right Triangles Hypotenuse Θ

4 Trigonometric Functions Definition Let θ be the acute angle of a right triangle and let opp, adj, and hyp be the lengths of the opposite, adjacent, and hypotenuse sides respectively of the right triangle. The six trigonometric functions of θ are defined as below. sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp

5 Example Find the values of the six trigonometric functions for the acute angle in the right triangle below. 3 Θ 6

6 Example Find the values of the six trigonometric functions for the acute angle in the right triangle below. 3 Θ 6 hyp = = 45 = 3 5

7 Example Find the values of the six trigonometric functions for the acute angle in the right triangle below. 3 Θ 6 hyp = = 45 = 3 5 sin θ = 3 3 = csc θ = 5 cos θ = 6 sec θ = 3 = tan θ = 3 6 = 1 2 cot θ = 2

8 Example Construct a right triangle with acute angle θ such that 3 cot θ = 3.

9 Example Construct a right triangle with acute angle θ such that 3 cot θ = Θ

10 Special Angles We frequently encounter the angles of measure 30 (π/6 radians), 45 (π/4 radians), and 60 (π/3 radians). We should know by heart the following trigonometric values. sin 30 = sin π 6 = 1 2 sin 45 = sin π 4 = 2 2 sin 60 = sin π 3 = 3 2 cos 30 = cos π 6 = 3 2 cos 45 = cos π 4 = 2 2 tan 30 = tan π 6 = 3 3 tan 45 = tan π 4 = 1 tan 60 = tan π 3 = 3 cos 60 = cos π 3 = 1 2

11 Cofunctions Recall: complementary angles have a sum equal to 90 (π/2 radians).

12 Cofunctions Recall: complementary angles have a sum equal to 90 (π/2 radians). Cosine is the cofunction of sine. Cotangent is the cofunction of tangent. Cosecant is the cofunction of secant. The cofunctions of complementary angles are equal.

13 Cofunctions Recall: complementary angles have a sum equal to 90 (π/2 radians). Cosine is the cofunction of sine. Cotangent is the cofunction of tangent. Cosecant is the cofunction of secant. The cofunctions of complementary angles are equal. sin(90 θ) = cos θ tan(90 θ) = cot θ sec(90 θ) = csc θ cos(90 θ) = sin θ cot(90 θ) = tan θ csc(90 θ) = sec θ

14 Fundamental Trigonometric Identities Reciprocal Identities sin θ = 1 csc θ csc θ = 1 sin θ Quotient Identities cos θ = 1 sec θ sec θ = 1 cos θ tan θ = 1 cot θ cot θ = 1 tan θ tan θ = sin θ cos θ cot θ = cos θ sin θ Pythagorean Identities sin 2 θ + cos 2 θ = tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ

15 Example If cos θ = 7 4 find the following trigonometric values. sec θ = sin θ = cot θ = sin(90 θ) =

16 Example 7 If cos θ = find the following trigonometric values. 4 sec θ = sin θ = cot θ = sin(90 θ) =

17 Example 7 If cos θ = find the following trigonometric values. 4 sec θ = sin θ = 3 4 cot θ = sin(90 θ) = cos θ = 7 4

18 Application A biologist wants to know the width w of a river so that instruments for studying the pollutants in the water can be set properly. From point A the biologist walks downstream 100 feet and sights to point C. From this sighting, it is determined that θ = 54. How wide is the river? C w Θ A 100

19 Solution Using the right triangle, tan θ = w 100 w = 100 tan θ = 100 tan feet

20 Homework Read Section 4.3. Exercises: 1, 5, 9, 13,..., 69, 73

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