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Goal: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. • Draw and find angles in standard position. • Convert between degree measures and radian measures. NEW VOCABULARY A curved arrow is drawn to the ending position of the ray, or terminal side of the angle. Angle of rotation: is an angle formed by the starting and ending positions of a ray. The angle is in standard position in a coordinate plane when the starting position of the ray, or initial side of the angle, is on the positive x-axis. Angle Rotations Positive Theta (between 0° − 360°) the arrow is drawn Counterclockwise Negative Theta (between 0° − 360°) the arrow is drawn clockwise Example: Standard Form A. Draw an angle with a measure of 210° in standard position. B. Draw an angle with a measure of –45° in standard position. Draw an Angle in Standard Position C. DIVING In a springboard diving competition, a diver made a 900-degree rotation before slicing into the water. Draw an angle in standard position that measures 900°. 900 = 2.5 360 Co-Terminal Angles: Twin Angles Co-terminal angles are angles that share the same terminal side. Can be negative or > 360° A) What is the positive co-terminal angle for 420° between [0,360]? 420 − 360 = 60° B) What is the negative co-terminal angle for 420°? between [0,-360]? 420 − 360 = 60 60 − 360 = −300° Your Turn! SNOWBOARDING While riding down the mountain, a snowboarder goes off a jump and turns 600° before touching down onto the snow again. Determine how many degrees past the positive x-axis the snowboarder lands. A. 120° B. 180° C. 240° D. 300° Your Turn! A. Draw an angle with a measure of 225° in standard position. C. A. B. D. B. Draw an angle with a measure of –60° in standard position. A. B. C. D. Your Turn! A. Find an angle with a positive measure and an angle with a negative measure that are co-terminal with 330°. A. –30°, 690° C. –60°, 630° B. –30°, 630° D. –60°, 720° B. Find an angle with a positive measure and an angle with a negative measure that are co-terminal with –80°. What is a Radian? • The radian measure of the angle is the 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ ratio 𝜃 = . We use radius=1 𝑟𝑎𝑑𝑖𝑢𝑠 • Recall that there are 360° in a full circle and the circumference is 2𝜋. • Therefore 360° = 2𝜋 1 = 2𝜋 Radian to Degrees to Radians Convert Between Degrees and Radians A. Rewrite 30° in radians. B. Rewrite 𝜋 𝜋 30° ∙ = 180° 6 in degrees. −5𝜋 180° ∙ = −300° 3 𝜋 C. Complete the Table 22.5° 180° 𝜋 240 ° Your Turn! A. Rewrite 45° in radians. A. C. B. D. C. Rewrite -495° in radians. B. Rewrite D. Rewrite 13𝜋 12 in degrees. in degrees. The Unit Circle 𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏 The unit circle below shows the measures of angles of rotation that are commonly used in trigonometry, with radian measures outside the circle and degree measures inside the circle. Provide the missing measures. Your Turn! The Unit Circle 𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏 The Unit Circle 𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏 𝑃𝑟𝑖𝑜𝑟 𝐾𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒 𝑁𝑒𝑒𝑑𝑒𝑑: 𝑺𝑶𝑯 − 𝑪𝑨𝑯 − (𝑻𝑶𝑨) 𝑂𝑝𝑝 = 𝐻𝑦𝑝 𝐴𝑑𝑗 = 𝐻𝑦𝑝 𝐼𝑓 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑠 1: 𝐹𝑜𝑟 𝐴𝑁𝑌 𝑟𝑎𝑑𝑖𝑢𝑠: ( x ,y ) 𝑆𝑖𝑛 = 𝑦 𝐶𝑜𝑠 = 𝑥 𝑦 𝑆𝑖𝑛 = 𝑟 x 𝐶𝑜𝑠 = r Class Activity: • Finding the 𝑠𝑖𝑛𝑒 𝜃, 𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 of each angle in each quadrant for the special triangles • I will: do the first one. • As a class: we do the second • In pairs: do the third 𝟐 (− 𝟐, 𝟐 𝟐) ( 𝟐 𝟐, 𝟐 45-45-90 Triangle 𝟐) 1) Inscribe the 45-45-90 Triangle in all four quadrants. 2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃 sin 𝜃 = (− 𝟐 𝟐, − 𝟐 𝟐) ( 𝟐 𝟐, − 𝟐 𝑂𝑝𝑝 𝑦 = 𝐻𝑦𝑝 1 𝐴𝑑𝑗 𝑥 𝑐𝑜𝑠𝜃 = = 𝐻𝑦𝑝 1 𝟐) 135 225 2 − 22 − 22 2 − 22 315 − 22 2 2 (− 𝟏 𝟐, 𝟑 𝟐) (𝟏 𝟐, 𝟑 60-30-90 Triangle 𝟐) 1) Inscribe the 60-30-90 Triangle in all four quadrants. 2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃 sin 𝜃 = (− 𝟏 𝟐, − 𝟑 𝟐) (𝟏 𝟐, − 𝟑 𝑂𝑝𝑝 𝑦 = 𝐻𝑦𝑝 1 𝐴𝑑𝑗 𝑥 𝑐𝑜𝑠𝜃 = = 𝐻𝑦𝑝 1 𝟐) 120 240 300 3 − 32 −1 2 − 32 1 2 2 −1 2 30-60-90 Triangle 1) Inscribe the 30-60-90 Triangle in all four quadrants. 2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃 . sin 𝜃 = 𝑂𝑝𝑝 𝑦 = 𝐻𝑦𝑝 1 150 1 2 − 32 𝐴𝑑𝑗 𝑥 𝑐𝑜𝑠𝜃 = = 𝐻𝑦𝑝 1 210 −1 2 − 3 2 330 −1 2 3 2 You can now be able to tell me the sine and cosine of any angle OR Point in the circle or beyond the circle! MEMORIZE OR KNOW HOW TO SOLVE Watch: https://learnzillion.com/lesson_plans/8031-understand-the-wrapping-functionusing-the-unit-circle 0.29 s - end Other things you can do: • What if I give you a Triangle? • What if I give you a point? Find sin , cos 𝑜𝑝𝑝 4 12 𝑠𝑖𝑛𝜃 = ℎ𝑦𝑝 = 15 = 5 9 3 𝑎𝑑𝑗 = = 𝑐𝑜𝑠𝜃 = 15 5 ℎ𝑦𝑝 15 The terminal side of angle in standard position intersects the unit circle at 3 𝑠𝑖𝑛𝜃 = 𝑦 = 4 7 𝑐𝑜𝑠𝜃 = 𝑥 = 4 .Find cos and sin . Your Turn! The terminal side of angle in standard position intersects the unit circle at Find cos and sin . The reference angle of a rotation angle is the acute angle from the terminal side of the rotation angle to the x-axis. (+, +) (−, +) (𝑪𝒐𝒔, 𝑺𝒊𝒏) (−, −) RECALL: (+, −) 𝐶𝑜𝑠 = 𝑥 𝑆𝑖𝑛 = 𝑦 1. Find the Co-terminal Angle 2. Use table to find Reference Angle 3. Choose the sign (positive or negative) based on Quadrant (or table) Sin 7𝜋 1. Divide the number to find how many half rotations 7 1 3 full rotations (2𝜋) +1 half rotation (𝜋) Where is 7𝜋 on the circle? Edge of the circle 𝑵𝑶𝑻𝑬: 𝝅 𝒊𝒔 𝒂 𝒉𝒂𝒍𝒇 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏 (−1,0) 5𝜋 𝜋 7𝜋 3𝜋 What is this coordinate point? Therefore, sin(7𝜋) = 0 6𝜋 0 4𝜋 2𝜋 Sin 7𝜋 1. Divide the number to find how many half rotations 7 1 3 full rotations (2𝜋) +1 half rotation (𝜋) Edge of the circle 2. Rewrite the question without the full rotations 1 half rotation = 𝜋 si𝑛 𝜋 = 0 1. Divide the number to find how many rotations 2 full rotations (2𝜋) 16 1 =5 +1 half rotation (𝜋) 3 3 + 1/3 2. Rewrite the question without the full rotations 4𝜋 𝜋 + 1/3 𝜋 = 3 4𝜋 cos 3 Third quadrant 𝑵𝑶𝑻𝑬: 𝝅 𝒊𝒔 𝒂 𝒉𝒂𝒍𝒇 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏 𝜋 5𝜋 3𝜋 What is this coordinate point? 1 3 − ,− , 2 2 60° 4𝜋 3 Therefore, cos(16𝜋/3) = −1/2 0 4𝜋 2𝜋 4𝜋 cos 3 3. Use Reference table (if necessary) 𝜋 180° 4𝜋 𝜃′ = −𝜋 = × = 60° 3 3 𝜋 cos 𝜃′ = 1/2 **Cos is negative in the third quadrant** 16𝜋 1 cos =− 3 2 Extra Practice with Co-terminal and Evaluating Sine and Cosine Calculator: Sine and Cosine 1)Press the number 2) Click sin or cos To find cos 3 3 To find sin 3 ∗𝜋 = 2 = −.98999 3 𝜋 2 = −1 Introducing your new FAVORITE Pythagorean Theorem! Favorite Identity: The Proof The terminal side of an angle θ intersects the unit circle at the point (a, b) as shown. Write a and b in terms of trigonometric functions involving θ. cos 𝜃 a= ________ sin 𝜃 b=________ cos 𝜃 cos 2 𝜃 sin 𝜃 sin2 𝜃 1 1 Favorite Identity: Confirming it works 2 2 sin 𝜃 + cos 𝜃 = 1 𝜋 sin 3 3 2 2 2 𝜋 + cos 3 2 2 =1 1 + =1 2 3 1 + =1 4 4 1. Plug in given value of sin or cosine 2. Solve for the missing trig function 3. Find the quadrant of the given Trig Function. Use this to decide if and should be positive or negative sin2 𝜃 + cos2 𝜃 = 1 0.766 2 (+, +) + cos2 𝜃 = 1 0.586 + cos2 𝜃 = 1 cos2 𝜃 = 1 − 0.586 cos2 𝜃 = 0.414 cos2 𝜃 = 0.414 cos 𝜃 = ± 0.643 Cos is positive in first quadrant 𝑊𝐴𝐼𝑇! 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒? sin2 𝜃 + cos2 𝜃 = 1 sin2 𝜃 + −0.906 2 =1 sin2 𝜃 + 0.821 = 1 (−, +) sin2 𝜃 = 1 − 0.821 sin2 𝜃 = 0.179 sin2 𝜃 = 0.179 sin 𝜃 = ±0.423 Sin is positive in first quadrant 𝑊𝐴𝐼𝑇! 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒? Your Turn! A. Find the exact value of cos 660°. B. Find the exact 7𝜋 value of sin(− ) 4 1. Find the exact value of sin 225°. A. B. C. D.