Evaluating Trig Functions Of Any Angle TUTORIAL Click the speaker icon on each slide to hear the narration

First Concept: Evaluating a trig function of a special angle 1.Sketch the angle in standard position 2.Determine the reference angle 3.Draw the triangle showing x, y, r with their values, based on the side ratios of the reference angle 4.Take the appropriate ratio of the sides 5.Simplify your ratio, if necessary 1.Sketch the angle in standard position 2.Determine the reference angle 3.Draw the triangle showing x, y, r with their values, based on the side ratios of the reference angle 4.Take the appropriate ratio of the sides 5.Simplify your ratio, if necessary

Example:  Sketch the terminal side of the angle in standard position

Example:  Find the reference angle:

Example:  Label the values of x, y, and r, paying close attention to the signs (r is always positive): 60  11 2

Example:  Compute the tangent ratio: 60  11 2

Second Concept: Trig functions of angles that lie on the axes § Trig functions of 90 , 180 , 270 , and 360  can be tricky § Steps: 1.Draw the angle and indicate x, y, and r 2.Use the following definitions: § Trig functions of 90 , 180 , 270 , and 360  can be tricky § Steps: 1.Draw the angle and indicate x, y, and r 2.Use the following definitions:

Example: Functions of 270   Draw the angle in standard position 270 

Example: Functions of 270   Indicate the coordinates of the endpoint of the terminal ray (always make r = 1) x = 0 y = –1 r = 1

Example: Functions of 270   Take the appropriate ratios to compute sin, cos, and tan x = 0 y = –1 r = 1

Third Concept: Angles with similar ratios § Every angle with the same reference angle will have a similar ratio § Identical to each other, or… § Different sign from each other § Use knowledge of the quadrants and x, y, r to know whether the ratio is positive or negative in that quadrant § r is always positive § Every angle with the same reference angle will have a similar ratio § Identical to each other, or… § Different sign from each other § Use knowledge of the quadrants and x, y, r to know whether the ratio is positive or negative in that quadrant § r is always positive

Signs of functions in each quadrant  sin  = y/r, so sin is positive where y is positive (Quadrants 1 and 2) and negative where y is negative (Quadrants 3 and 4)  cos  = x/r, so cos is positive where x is positive (Quadrants 1 and 4) and negative where x is negative (Quadrants 2 and 3)  tan  = y/x, so tan is positive where x and y have the same sign (Quadrants 1 and 3) and negative where x and y have different signs (Quadrants 2 and 4)  sin  = y/r, so sin is positive where y is positive (Quadrants 1 and 2) and negative where y is negative (Quadrants 3 and 4)  cos  = x/r, so cos is positive where x is positive (Quadrants 1 and 4) and negative where x is negative (Quadrants 2 and 3)  tan  = y/x, so tan is positive where x and y have the same sign (Quadrants 1 and 3) and negative where x and y have different signs (Quadrants 2 and 4)

Summary chart A ll trig functions are positive S in is positive, others are negative T an is positive, others are negative C os is positive, others are negative x and y are positive x is neg y is pos x and y are negative x is pos y is neg AS TC “All Students Take Calculus” “All Schools Torture Children” “Avoid Silly Trig Classes” Mnemonic:

Example: cos 35  =  Other angles in the family (meaning they have a reference angle equal to 35  )  In the second quadrant, 145  has the same reference angle and the cosine is negative, so cos 145  = –0.819  In the third quadrant, 215  has the same reference angle and the cosine is negative, so cos 215  = –0.819  In the fourth quadrant, 325  has the same reference angle and the cosine is positive, so cos 325  =  Other angles in the family (meaning they have a reference angle equal to 35  )  In the second quadrant, 145  has the same reference angle and the cosine is negative, so cos 145  = –0.819  In the third quadrant, 215  has the same reference angle and the cosine is negative, so cos 215  = –0.819  In the fourth quadrant, 325  has the same reference angle and the cosine is positive, so cos 325  = 0.819

The End  Hope you enjoyed the show!