1 / 18

Trigonometric Functions Evaluation at Point

This example demonstrates how to evaluate the trigonometric functions of an angle at a given point using the Pythagorean theorem and the unit circle.

gordonsmith
Download Presentation

Trigonometric Functions Evaluation at Point

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ. r √ 25 x2 + y2 (–4)2 + 32 √ √ = = = EXAMPLE 1 Evaluate trigonometric functions given a point SOLUTION Use the Pythagorean theorem to find the value of r. = 5

  2. 5 4 4 3 5 3 y x – sinθ cosθ = = = = 4 3 3 4 5 5 r r y r – tanθ cscθ = = = = x y r x – secθ – cotθ = = = = x y EXAMPLE 1 Evaluate trigonometric functions given a point Using x = –4, y = 3, and r =5, you can write the following:

  3. Use the unit circle to evaluate the six trigonometric functions of = 270°. θ Draw the unit circle, then draw the angle θ = 270° in standard position. The terminal side of θ intersects the unit circle at (0, –1), so use x=0 and y= –1 to evaluate the trigonometric functions. EXAMPLE 2 Use the unit circle SOLUTION

  4. 1 0 r x secθ cosθ = = = = 0 1 x r –1 0 y x tanθ cotθ = = = = 0 –1 x y 1 1 – y r sinθ cscθ = = = = – 1 1 r y EXAMPLE 2 Use the unit circle = –1 = –1 undefined = 0 undefined = 0

  5. 5π 5π π Find the reference angle θ' for (a) θ= 3 3 3 and (b) θ = – 130°. a. The terminal side of θ lies in Quadrant IV. So, θ' = 2π– . = b. Note that θ is coterminal with 230°, whose terminal side lies in Quadrant III. So, θ' = 230° –180° + 50°. EXAMPLE 3 Find reference angles SOLUTION

  6. 17π Evaluate (a) tan ( – 240°) and (b) csc . 6 a. The angle –240° is coterminal with 120°. The reference angle is θ' = 180° – 120° = 60°. The tangent function is negative in Quadrant II, so you can write: 3 √ tan (–240°) = – tan 60° = – EXAMPLE 4 Use reference angles to evaluate functions SOLUTION

  7. 17π 17π b. The angle is coterminal with . The reference angle is θ' = π – = . The cosecant function is positive in Quadrant II, so you can write: 6 6 5π 5π 5π csc = csc = 2 π 6 6 6 6 EXAMPLE 4 Use reference angles to evaluate functions

  8. sin 2θ d = v2 32 EXAMPLE 5 Calculate horizontal distance traveled Robotics The “frogbot” is a robot designed for exploring rough terrain on other planets. It can jump at a 45° angle and with an initial speed of 16 feet per second. On Earth, the horizontal distance d(in feet) traveled by a projectile launched at an angle θ and with an initial speed v(in feet per second) is given by: How far can the frogbot jump on Earth?

  9. 162 sin (2 45°) d = 32 sin 2θ d = v2 32 EXAMPLE 5 Calculate horizontal distance traveled SOLUTION Write model for horizontal distance. Substitute 16 for vand 45° for θ. = 8 Simplify. The frogbot can jump a horizontal distance of 8 feet on Earth.

  10. √ √ √ √ 3 3 3 3 a. cos–1 2 2 2 2 When0θπor0°180°,the angle whose cosine is ≤ ≤ ≤ θ ≤ a. π 30° cos–1 = θ = 6 cos–1 = θ = EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. SOLUTION

  11. b. 2 sin–1 There is no angle whose sine is 2. So, is undefined. b. sin–1 2 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. SOLUTION

  12. ( – ) √ √ √ √ c. tan–1 3 3 c. When –< θ < , or – 90°< θ < 90°, the angle whose tangent is – is: – ( – ) ( – ) –60° = = θ tan–1 3 θ tan–1 3 = = π π π 2 3 2 EXAMPLE 1 Evaluate inverse trigonometric functions Evaluate the expression in both radians and degrees. SOLUTION

  13. Solve the equationsinθ = – where180° < θ < 270°. ≤ ≤ interval –90° θ 90°, the angle whose Use a calculator to determine that in the 5 5 5 8 8 8 – sine is – is sin–1– 38.7°. This angle is in Quadrant IV, as shown. EXAMPLE 2 Solve a trigonometric equation SOLUTION STEP 1

  14. Find the angle in Quadrant III (where 180° < θ < 270°) that has the same sine value as the angle in Step 1. The angle is: 180° + 38.7° = 218.7° θ 5 –  = sin 218.7° – 0.625 8 EXAMPLE 2 Solve a trigonometric equation STEP 2 Use a calculator to check the answer. CHECK :

  15. The correct answer is C. ANSWER 6 6 adj 56.9° θ cos – 1 = cosθ = = 11 11 hyp EXAMPLE 3 Standardized Test Practice SOLUTION In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.

  16. Monster Trucks EXAMPLE 4 Write and solve a trigonometric equation A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?

  17. Draw: a triangle that represents the ramp. 8 opp tanθ = = 20 adj EXAMPLE 4 Write and solve a trigonometric equation SOLUTION STEP 1 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length. STEP 2

  18. ANSWER The angle of the ramp is about 22°. 8 21.8° θ tan–1 = 20 EXAMPLE 4 Write and solve a trigonometric equation STEP 3 Use: a calculator to find the measure of θ.

More Related