T7.2 Right Triangle Trigonometry Presentation
- 1. T7.2 Right Triangle Trigonometry Chapter 7 The Unit Circle: Sine and Cosine Functions
- 2. Concepts and Objectives ⚫ The objectives for this section are ⚫ Use right triangles to evaluate trigonometric functions ⚫ Use equal cofunctions of complementary angles ⚫ Use the definitions of trigonometric functions for any angle ⚫ Use right-triangle trigonometry to solve applied problems
- 3. Trigonometric Ratio Review ⚫ In Geometry, we learned that for any given right triangle, there are special ratios between the sides. A opposite adjacent = opposite sin hypotenuse A = adjacent cos hypotenuse A = opposite tan adjacent A
- 4. Trigonometric Functions ⚫ Consider a circle centered at the origin with radius r: ⚫ The equation for this circle is x2 + y2 = r2 ⚫ A point (x, y) on the circle creates a right triangle whose sides are x, y, and r. ⚫ The trig ratios are now (x, y) r x y = sin y r = cos x r = tan y x
- 5. Trigonometric Functions ⚫ There are three other ratios in addition to the three we already know : cosecant, secant, and cotangent. ⚫ These ratios are the inverses of the original three: (x, y) r x y = = 1 csc sin r y = = 1 sec cos r x = = 1 cot tan x y
- 6. Finding Function Values ⚫ Example: The terminal side of an angle in standard position passes through the point (15, 8). Find the values of the six trigonometric functions of angle . (15, 8)
- 7. Finding Function Values ⚫ Example: The terminal side of an angle in standard position passes through the point (15, 8). Find the values of the six trigonometric functions of angle . 8 15 (15, 8) We know that x = 15 and y = 8, but we still have to calculate r: Now, we can calculate the values. = + 2 2 r x y = + = 2 2 15 8 17 17
- 8. Finding Function Values ⚫ Example: The terminal side of an angle in standard position passes through the point (15, 8). Find the values of the six trigonometric functions of angle . 8 15 (15, 8) 17 = = 8 sin 17 y r = = 15 cos 17 x r = = 8 tan 15 y x = = 17 csc 8 r y = = 17 sec 15 r x = = 15 cot 8 x y
- 9. The Unit Circle ⚫ Angles in standard position whose terminal sides lie on the x-axis or y-axis (90°, 180°, 270°, etc.) are called quadrantal angles. ⚫ To find function values of quandrantal angles easily, we ⚫ Notice that at the quadrantal angle points x and y are either 0, 1, or –1 (r is always 1). use a circle with a radius of 1, which is called a unit circle. 90 (0, 1) (0, –1) 270 180 (–1, 0) 0/360 (1, 0)
- 10. Values of Quadrantal Angles ⚫ Example: Find the values of the six trigonometric functions for an angle of 270°.
- 11. Values of Quadrantal Angles ⚫ Example: Find the values of the six trigonometric functions for an angle of 270°. At 270°, x = 0, y = –1, r = 1. − = = − 1 sin270 1 1 = = 0 cos270 0 1 − = = 1 tan270 undefined 0 (0, –1)
- 12. Values of Quadrantal Angles ⚫ Example: Find the values of the six trigonometric functions for an angle of 270°. At 270°, x = 0, y = –1, r = 1. = = − − 1 csc270 1 1 = = 1 sec270 undefined 0 = = − 0 cot270 0 1 (0, –1)
- 13. Identifying an Angle’s Quadrant ⚫ To identify the quadrant of an angle given certain conditions, note the following: ⚫ In the first quadrant, x and y are both positive. ⚫ In QII, x is negative and y is positive. ⚫ In QIII, both are negative. ⚫ In QIV, x is positive and y is IV III II I (+,+) (–,+) (–,–) negative. (+,–)
- 14. Identifying an Angle’s Quadrant ⚫ Example: Identify the quadrant (or possible quadrants) of an angle that satisfies the given conditions. a) sin > 0, tan < 0 b) cos < 0, sec < 0
- 15. Identifying an Angle’s Quadrant ⚫ Example: Identify the quadrant (or possible quadrants) of an angle that satisfies the given conditions. a) sin > 0, tan < 0 b) cos < 0, sec < 0 I, II II, IV II II, III II, III II, III
- 16. Classwork ⚫ Algebra & Trigonometry 2e ⚫ 7.2: 10-22 (even); 7.1: 40-48 (even) ⚫ College Algebra 2e ⚫ 9.6: 30-38 (even) ⚫ T7.2 Classwork Check ⚫ Quiz T7.1