Section 12 1 Trigonometric Functions in Right Triangles

Section 12. 1: Trigonometric Functions in Right Triangles

Theta the Greek letter that is often used to represent the measure of an acute angle in a right triangle. Trigonometric compares the side lengths of a right triangle. Ratio Trigonometric Functions

the leg ACROSS from the Opposite Side angle being used. of an Angle A the leg forming the angle Adjacent Side being used. of an Angle Hypotenuse the side ACROSS from the right angle. C θ B

Trigonometric Functions in Right Triangles SOH CAH TOA Sine Cosine Tangent Symbols Examples

SOH CAH TOA Cosecant Secant Cotangent Symbols Examples

Example 1: Find the values of the six trigonometric functions for angle G.

Example 2: a) Find Cos A. B 13 5 b) Find Tan B. C 12 A

Example 3: In a right triangle, B is acute and cos B = of tan B. Find the value

Example 4: If find the exact values of the five remaining trigonometric functions for A. B C A

Example 5: If find the exact values of the five remaining trigonometric functions for A. B C A

Example 6: Find the values of the six trigonometric functions for angle θ. a)

Example 6: Find the values of the six trigonometric functions for angle θ. b)

Example 7: Given the triangle, find sec A. B 13 5 C 12 A

Side a will always be across from A B Side b will always be across from B Side c will always be across from C Relationships between Sides and Angles A c b a C

Example 8: In a right triangle, A and B are acute. a) If tan A = what is cos A?

Example 8 cont. : In a right triangle, A and B are acute. a) If cos A = what is tan A?

Example 9: Use a trigonometric function to find the value of x.

Example 10: Use a trigonometric function to find the value of x. 30° x 5

Example 11: Use a trigonometric function to find the value of x. x 47° 12

Words: If A is an acute angle and the sine of A is x, then the inverse sine of x is the measure of A. Symbols: If sin A = x, then sin-1 x = m A. Inverse Trigonometric Words: If A is an acute angle and the cosine of A is x, then the Functions inverse cosine of x is the measure of A. Symbols: If cos A = x, then cos-1 x = m A.

Words: If A is an acute angle and the tangent of A is x, then the inverse tangent of x is the measure of A. Inverse -1 Trigonometric Symbols: If tan A = x, then tan x = m A. Functions

Example 12: a) Find the measure of � B. Round to the nearest tenth if necessary.

Example 12 cont. : b) Find the measure of � A. Round to the nearest tenth if necessary.

Example 13: Find the measure of � A. Round to the nearest tenth if necessary.

Example 14: Find the measure of θ in the following triangles. a)

Example 14 cont. : Find the measure of θ in the following triangles. b)

Example 14 cont. : Find the measure of θ in the following triangles. c)

Example 15: Solve ΔABC. B 14 A b 6 C

Angle of Elevation The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

Angle of Depression The angle below horizontal that an observer must look to see an object that is lower than the observer.

Example 16: To calculate the height of a tree in his front yard, Anand walked 50 feet from the base of the tree and used an inclinometer to measure the angle from his eye to the top of the tree, which was 62°. If Anand’s eye level is at 6 feet, about how tall is the tree?

Example 17: To calculate the height of a building, Joel walked 200 feet from the base of the building and used an inclinometer to measure the angle from his eye to the top of the building. If Joel’s eye level is at 6 feet, what is the distance from the top of the building to Joel’s eye?

Example 18: A golfer is standing at the tee, looking up to the green on a hill. The tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12°. From a camera in a blimp, the apparent distance between the golfer and the hole is the horizontal distance. Find the horizontal distance.

Example 19: When Baby J was a toddler, she was 30” tall. She often liked to play in the sandbox and look down at tiny ants crawling around. If her angle of depression to an ant is 39°, how far away are her feet from the ant? Round your answer to the nearest tenth of an inch.

Example 20: The recommended angle of elevation for a ladder used in firefighting is 75°. At what height on a building does a 21 -foot ladder reach if the recommended angle of elevation is used? Round to the nearest tenth.

Example 21: Mario hits a line drive home run from 3 feet in the air to a height of 125 feet, where it strikes a billboard in the outfield. If the angle of elevation of the hit was 22°, what is the horizontal distance from home plate to the billboard?

When do you use Trigonometric functions? When do you use Inverse Trigonometric Functions?