Trigonometry for Any Angle Pre Calculus Trigonometry for Any Angle

Right Triangle Trigonometry

Fill out some stuff we already know Sine Cosine Tangent Cosecant Secant Cotangent Abbreviation Reciprocal Function Co-function Right Triangle Definition Unit Circle Definition Any Angle Definition Positive Quadrants Negative Quadrants Odd or Even Domain Range Period Inverse Inverse Domain

Fill out some stuff we already know Sine Cosine Tangent Cosecant Secant Cotangent Abbreviation sin cos tan csc sec cot Reciprocal Function Co-function Right Triangle Definition Unit Circle Definition Any Angle Definition Positive Quadrants Negative Quadrants Odd or Even

Fill out some stuff we already know Sine Cosine Tangent Cosecant Secant Cotangent Abbreviation sin cos tan csc sec cot Reciprocal Function Co-function Right Triangle Definition Opp/hyp Adj/hyp Opp/adj Hyp/opp Hyp/adj Adj/opp Unit Circle Definition y x y/x 1/ y 1/x x/y Any Angle Definition Positive Quadrants Negative Quadrants Odd or Even

Fill out some stuff we already know Sine Cosine Tangent Cosecant Secant Cotangent Abbreviation sin cos tan csc sec cot Reciprocal Function Co-function Right Triangle Definition Opp/hyp Adj/hyp Opp/adj Hyp/opp Hyp/adj Adj/opp Unit Circle Definition y x y/x 1/ y 1/x x/y Any Angle Definition Positive Quadrants 1 and 2 1 and 4 1 and 3 Negative Quadrants 3 and 4 2 and 3 2 and 4 Odd or Even

Even and Odd Trig Functions An even function: f(x) = f(-x) cos(30o) = cos(-30o)? cos(135o) = cos(-135o)? The cosine and its reciprocal are even functions.

Even and Odd Trig Functions An odd function: f(-x) = -f(x) sin(-30o) = -sin(30o)? sin(-135o) = -sin(135o)? The sine and its reciprocal are odd functions.

Even and Odd Trig Functions An odd function: f(-x) = -f(x) tan(-30o) = -tan(30o)? tan(-135o) = -tan(135o)? The tangent and its reciprocal are odd functions.

Even and Odd Trig Functions Cosine and secant functions are even cos (-t) = cos t sec (-t) = sec t Sine, cosecant, tangent and cotangent are odd sin (-t) = - sin t csc (-t) = - csc t tan (-t) = - tan t cot (-t) = - cot t Add these to your worksheet

Unit Circle Review How can we memorize it? Symmetry For Radians the denominators help! Knowing the quadrant gives the correct + / - sign

Practice… Get out your Unit Circle, Pencil and Paper! ON YOUR OWN try these… Write the question and the answer

Check your work

How did you do??

Reference Angles Let θ be an angle in standard position. Its reference angle is the acute angle θ’ (called “theta prime”) formed by the terminal side of θ and the horizontal axis.

Look at these

Reference Angles Let θ be an angle in standard position and its reference angle has the same absolute value for the functions, the sign ( +/ - ) must be determined by the quadrant of the angle. Quadrant II θ’ = π – θ (radians) = 180o – θ (degrees) Quadrant III θ’ = θ – π (radians) = θ – 180o (degrees) Quadrant IV θ’ = 2π – θ (radians) = 360o – θ (degrees)

Trigonometry for any angle Given a point on the terminal side Let  be an angle in standard position with (x, y) a point on the terminal side of  and r be the length of the segment from the origin to the point r θ (x,y) Then….

Trig for any angle Add these definitions to summary worksheet The six trigonometric functions can be defined as Add these definitions to summary worksheet

Your Chart should look like… Sine Cosine Tangent Cosecant Secant Cotangent Abbreviation sin cos tan csc sec cot Reciprocal Function Co-function Right Triangle Definition Opp/hyp Adj/hyp Opp/adj Hyp/opp Hyp/adj Adj/opp Unit Circle Definition y x y/x 1/ y 1/x x/y Any Angle Definition y/r x/r r/y r/x Positive Quadrants 1 and 2 1 and 4 1 and 3 Negative Quadrants 3 and 4 2 and 3 2 and 4 Odd or Even Odd Even

Evaluating Trig Functions: Find sin, cos and tan given (-3, 4) is a point on the terminal side of an angle. Find r Find the ratio of the sides of the reference angle Make sure you have the correct sign based upon quadrant θ r (-3, 4) Find r. (-3)2 + (4)2= r2 r =5 sin θ = 4/5 cos θ = -3/5 tan θ = -4/3

Name the quadrant… A little different twist… The cosine and sine of the angle are positive 1 The cosine and sine are negative 3 The cosine is positive and the sine is negative. 4 The sine is positive and the tangent is negative 2 The tangent is positive and the cosine is negative. The secant is positive and the sine is negative.

Look at this one… Given tan  = -5/4 and the cos  > 0, find the sin  and sec . Which quadrant is it in? θ r (4, -5) The tangent is negative, and the cosine is positive Quadrant IV at point (4, -5) Find r and use the triangle to find the sine and secant

And this one… Let  be an angle in quadrant II such that sin  = 1/3 find the cos  and the tan . θ 3 (x, 1) Set up a triangle based upon the information given . Calculate the other side Find the other trigonometric functions