Trig Functions of Any Angle Section 4.4 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Review Determine the Trigonometric Functions for 𝜽 5 4 3 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

Equation in Standard Form For 𝜽 be an angle in standard position with any point 𝒙, 𝒚 𝐬𝐢𝐧 𝜽 = y/r 𝐜𝐨𝐬 𝜽 = x/r 𝐭𝐚𝐧 𝜽= y/x 𝐜𝐬𝐜 𝜽 = r/y 𝐬𝐞𝐜 𝜽 = r/x 𝐜𝐨𝐭 𝜽= x/y B. To establish the radius, the equation is 𝒓= 𝒙 𝟐 + 𝒚 𝟐 C. Think of “ASTC: All Students Take Calculus” A: All points are always positive in Quadrant I S: Sine points are positive in Quadrant II T: Tan points are positive in Quadrant III C: Cosine points are positive in Quadrant IV 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

Equation in Standard Form For 𝜽 be an angle in standard position with any point 𝒙, 𝒚 : S A When ALL trig functions are positive When SIN is positive Quadrant II (– , +) Quadrant I (+, +) T C When TAN is positive When COS is positive Quadrant III (–, –) Quadrant IV (+, –) “All Students Take Calculus” 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

Steps in Evaluating Functions given a Point Draw a picture from a coordinate plane Identify and plot the point onto the coordinate plane Determine the missing side using the radius equation Use Trigonometric Functions to solve 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 1 Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 4 3 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 1 Let 𝟑, 𝟒 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 3 4 5 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 2 Let 𝟏𝟎 𝟏𝟎 ,− 𝟑 𝟏𝟎 𝟏𝟎 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Your Turn Let 𝟏,−𝟏 be a point on the terminal side of 𝜽. Determine the value of the six trigonometric functions for 𝜽. 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 3 Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽= 𝟏 𝟑 , determine the value of the six trigonometric functions for 𝜽. 3 1 x 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 3 Let 𝜽 be in Quadrant II. Given 𝐬𝐢𝐧 𝜽= 𝟏 𝟑 , determine the value of the six trigonometric functions for 𝜽. 1 3 x 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 4 Let 𝐜𝐨𝐭 𝜽<𝟎. Given 𝐜𝐬𝐜 𝜽=𝟒 , determine the value of the six trigonometric functions for 𝜽. 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Example 5 Given 𝒚= 𝟏 𝟑 𝒙 and 𝜽 is in Quadrant III, determine the value of the six trigonometric functions for 𝜽. -3 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= -1 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Your Turn Let 𝐭𝐚𝐧 𝜽>𝟎. Given 𝐜𝐨𝐬 𝜽=− 𝟐 𝟑 , determine the value of the six trigonometric functions for 𝜽. 𝐬𝐢𝐧 𝜽= 𝐜𝐨𝐬 𝜽= 𝐭𝐚𝐧 𝜽= 𝐜𝐬𝐜 𝜽= 𝐬𝐞𝐜 𝜽= 𝐜𝐨𝐭 𝜽= 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle

§4.4: Trig Functions of Any Angle Assignment Page 296 11, 15-23 odd, 27-35 odd (omit 29) 5/20/2019 6:17 PM §4.4: Trig Functions of Any Angle