C H. 4 – T RIGONOMETRIC F UNCTIONS 4.2 – The Unit Circle

F UNDAMENTAL TRIG IDENTITIES Reciprocal Identities: Quotient Identities: Pythagorean Identities:

T HE UNIT CIRCLE The unit circle follows the equation x 2 + y 2 = 1 Radius = 1, center at the origin Angles always have the initial side on the positive x- axis Consider the functions of an angle in the first quadrant that intersects the circle at (x, y): cos( θ ) = x/1 = x sin( θ ) = y/1 = y tan( θ ) = y/x From these 3 functions, we get… sec(θ) = 1/x csc(θ) = 1/y cot(θ) = x/y (x, y)

S PECIAL RIGHT TRIANGLES Recall the dimensions of a and a right triangle: These relationships give us trigonometric values for common angles! x x 2x x 60º 30º (x, y) At, (x, y) =

(-1, 0)(1, 0) (0, 1) (0, -1) 30 º 45 º 60 º 0º0º 330 º 300 º 270 º 240 º 210 º 180 º 150 º 120 º 90 º 315 º 225 º 135 º 0 11π/6 5π/3 4π/3 2π/3 π/3 π/43π/4 5π/47π/4 π/65π/6 7π/6 3π/2 π/2 π

P ROPERTIES OF TRIG FUNCTIONS Cosine and secant are even functions cos(-θ) = cos(θ) sec(-θ) = sec(θ) Sine, cosecant, tangent, and cotangent are odd functions sin(-θ) = -sin(θ) csc(-θ) = -csc(θ) tan(-θ) = -tan(θ) cot(-θ) = -cot(θ) Trig functions are periodic sin(θ + 2πn) = sin(θ) cos(θ + 2πn) = cos(θ) tan(θ + πn) = tan(θ)

To memorize the unit circle, know your reference angles!!! Ex: Find the 6 trigonometric functions at θ = 2π/3. Reference angle = π/3  Since 2π/3 is in quadrant II, the x is negative  Using this coordinate, we can find the 6 trig functions:

E VALUATE WITHOUT A CALCULATOR :

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