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The Trigonometric Functions What about angles greater than 90°? 180°? The trigonometric functions are defined in terms of a point on a terminal side r is found by using the Pythagorean Theorem: r x y 2 2 The 6 Trigonometric Functions of angle are: sin r csc , y 0 y cos r sec , x 0 x x cot , y 0 y tan , x0 sin y r The Trigonometric Functions The trigonometric values do not depend on the selected point – the ratios will be the same: First Quadrant: sin = + cos = + tan = + csc = + sec = + cot = + Second Quadrant: sin = + cos = tan = csc = + sec = cot = - Third Quadrant: sin = cos = tan = + csc = sec = cot = + y x Fourth Quadrant: sin = cos = + tan = csc = sec = + cot = - y x All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants: Star All Sine is positive All functions are positive Trig Tan is positive Class Cos is positive So, now we know the signs of the trig functions, but what about their values?... The value of any trig function of an angle is equal to the value of the corresponding trigonometric function of its reference angle, except possibly for the sign. The sign depends on the quadrant that is in. Reference Angles The reference angle, α, is the angle between the terminal side and the nearest x-axis: All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants: Star All Sine is positive All functions are positive Trig Tan is positive Class Cos is positive Quadrantal Angles (terminal side lies along an axis) Trig values of quadrantal angles: 0° 90° 180° 270° 360° sin 0 1 0 –1 0 cos 1 0 –1 0 1 tan 0 undefined 0 undefined 0 cot undefined 0 undefined 0 undefined sec 1 undefined –1 undefined 1 undefined 1 undefined –1 undefined csc Trigonometric Identities Reciprocal Identities 1 sin x csc x 1 cos x sec x 1 tan x cot x Quotient Identities sin x tan x cos x cos x cot x sin x Trigonometric Identities Pythagorean Identities The fundamental Pythagorean identity: sin 2 x cos 2 x 1 Divide the first by : 1 cot x csc x Divide the first by cos2x : tan 2 x 1 sec 2 x sin2x 2 2