"Sec 2 Trigonometric Functions"
Sec 2 Trigonometric Expressions Defn: If t is a real number that represents the distance traveled along the unit circle and P(x, y) is the terminal point on the unit circle, then each trigonometric expression is defined as follows: *the sine is defined to be sint = y *the cosine is defined to be cost = x *the tangent is defined to be tant = y/x *the cosecant is defined to be csct = 1/y *the secant is defined to be sect = 1/x *the cotangent is defined to be cott = x/y To evaluate a trigonometric expression: 1) Move the distance t on the unit circle 2) Locate the terminal point P associated with t 3) Use the definition and P(x,y) to find the value 4) The quadrant will determine the sign of the value, unless the point is on an axis. *REDUCE ALL RATIOS Ex: Find the exact value of each expression without using a calculator. a) sin 2 b) cos ( ) 2 3 c) tan 2 d) sec 0 e) cot ( ) 2 f) cos 3 g) tan ( ) 3 h) sin 19 4 i) cos ( ) 4 7 j) sec 6 11 k) cot 3 The value of trig expressions can also be found if only the terminal point is given. The distance will be represented as t. (REDUCE ALL RATIOS) Ex: Find the sint, cost, and tant by using the given terminal point determined by t. a) P ( 12 , 5 ) 13 13 5 61 6 61 b) P ( , ) 61 61 Ex: Find the quadrant in which the terminal point determined by t lies. a) sint < 0 and cost > 0 b) sint > 0 and tant > 0 c) sect < 0 and tant < 0 The calculator may be used to evaluate other trigonometric expressions. If t is given in terms of a real number then the mode must be set to the radian selection. Ex: Evaluate (set your calculator mode) a) sin 2.2 b) cos 1.1 c) cot 28 d) csc .98 Fundamental Identities of Trig Expressions: sin 1) tant = costt 2) cott = cost sin t 3) sin2t + cos2t = 1 4) tan2t + 1 = sec2t 5) 1 + cot2t = csc2t 6) csct = 1 sint 1 7) sect = cost 1 8) cott = tant Ex: Use the fundamental identities to evaluate the remaining trigonometric expressions. 3 a) cost = and t lands in Q4 5 b) cos t = 1 and t 3 3 2 3 c) csc t = -3 and t 2 2 4 d) cot t = and sin t > 0 3 *There are two ways to show if a function is even or odd: * If f(-x) = -f(x) then f(x) is odd. * If f(x) is swrt the origin then f(x) is odd. * If f(-x) = f(x) then f(x) is even. * If f(x) is swrt the y-axis then f(x) is even. The algebra test: 1) replace x with –x 2) simplify the expression 3) analyze the result. Summary of trig expressions: sin (-t) = -sin t csc (-t) = -csc t cos (-t) = cos t sec (-t) = sec t tan (-t) = -tan t cot (-t) = -cot t Ex: Determine if the function is even, odd or neither. a) f(x) = x2 cosx b) f(x) = x2 tanx