𝑥=4, 𝑦=5 𝑥=21, 𝑟=29 𝑟= 10 , 𝑥=1 𝑦=4 2 , 𝑥=4 Warm Up r y 𝒓= 𝟒𝟏 x 𝒚=𝟐𝟎 A right triangle has side lengths x, y, and r. Find the unknown length. r y 𝑥=4, 𝑦=5 𝑥=21, 𝑟=29 𝑟= 10 , 𝑥=1 𝑦=4 2 , 𝑥=4 The square has side lengths 14. The two curves are each 1 4 of a circle with radius 14. Find the area of the shaded region. 𝒓= 𝟒𝟏 x 𝒚=𝟐𝟎 𝒚=𝟑 𝒓=𝟒 𝟑 𝟗𝟖𝝅−𝟏𝟗𝟔
Quiz 7.1 & 7.2 Degree & Radian Conversions Coterminal Angles Arc Length of a Sector Area of a Sector Apparent Size
Section 7-3 The Sine and Cosine Functions Objective: To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.
𝑠𝑖𝑛𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑦 𝑟 r 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑥 𝑟 cos 𝜃=
Example 1 If the terminal ray of an angle θ in standard position passes through (-3, 2), find sin θ and cos θ. Solution: On a grid, locate (-3,2). Use this point to draw a right triangle, where one side is on the x-axis, and the hypotenuse is line segment between (-3,2) and (0,0). Start of Day 2
𝑟= −3 2 + 2 2 𝒙=−𝟑 𝒓= 𝟏𝟑 𝐲=𝟐 = 2 13 13 = 2 13 𝑠𝑖𝑛𝜃= 𝑦 𝑟 = −3 13 13 = 2 13 13 = 2 13 𝑠𝑖𝑛𝜃= 𝑦 𝑟 = −3 13 13 = −3 13 𝑐𝑜𝑠𝜃= 𝑥 𝑟 𝑟= −3 2 + 2 2 𝒙=−𝟑 𝒓= 𝟏𝟑 𝐲=𝟐
Example 2 If the 𝑠𝑖𝑛𝜃=− 5 13 , what quadrant is the angle in?
= 12 13 𝑐𝑜𝑠𝜃= 𝑥 𝑟 𝐲=−𝟓 𝒓=𝟏𝟑 𝑥= 13 2 − −5 2 𝒙=𝟏𝟐 𝒙=±𝟏𝟐 4th Quadrant, so
𝑠𝑖𝑛𝜃= cos 𝜃= 𝑦 𝑟 𝑥 𝑟 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑎𝑑𝑗 ℎ𝑦𝑝 = r 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑥 𝑟 cos 𝜃= When the radius =1 on the unit circle, 𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦 𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥
Unit Circle The circle x2 + y2 = 1 has radius 1 and is therefore called the unit circle. This circle is the easiest one with which to work because sin θ and cos θ are simply the y- and x-coordinates of the point where the terminal ray of θ intersects the circle. When the radius =1 on the unit circle, 𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦 𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥
1 1 2 1 2
II I III IV (−,+) (+,+) (−,−) (+,−) On Your Unit Circle: Label the quadrants. Note the positive or negative x and y values in each quadrant. (cos, sin) (cos, sin) (−,+) (+,+) II I III IV (−,−) (+,−) (cos, sin) (cos, sin)
You can determine the exact value of sine and cosine for many angles on the unit circle. 1 -1 − 3 2 2 2 Find: sin 90° sin 450° cos (-π) sin (− 2𝜋 3 ) cos -315° Refer to graph file “UC Quadrantal Angles”
Example 3 Degrees: 𝜃=90˚±360𝑛 Radians: 𝜃= 𝜋 2 ±2𝑛𝜋 Solve sin θ = 1 for θ in degrees and radians. Degrees: 𝜃=90˚±360𝑛 Radians: 𝜃= 𝜋 2 ±2𝑛𝜋
Repeating Sin and Cos Values For any integer n, 𝑠𝑖𝑛 (𝜃 ± 360°𝑛) = 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 (𝜃 ± 360°𝑛) =𝑐𝑜𝑠𝜃 𝑠𝑖𝑛 (𝜃 ±2𝜋𝑛) = 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 (𝜃 ±2𝜋𝑛) =𝑐𝑜𝑠𝜃 The sine and cosine functions are periodic. They have a fundamental period of 360˚ or 2 radians.
Homework Page 272 #1-27 odd, #33-41 odd