 # EXAMPLE 1 Find trigonometric values Given that sin  = and <  < π, find the values of the other five trigonometric functions of . 4 5 π 2.

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EXAMPLE 1 Find trigonometric values Given that sin  = and <  < π, find the values of the other five trigonometric functions of . 4 5 π 2

EXAMPLE 1 Find trigonometric values SOLUTION STEP 1 Find cos . Write Pythagorean Identity. sin  + cos  2 2 = 1 Substitute for sin . 4 5 ( ) + cos  4 5 2 2 1 = Subtract ( ) from each side. 4 5 2 cos  2 2 4 5 1 – ( ) = Simplify. cos  2 9 25 = Take square roots of each side. cos  3 5 + – = Because  is in Quadrant II, cos  is negative. cos  3 5 – =

EXAMPLE 1 Find trigonometric values STEP 2 Find the values of the other four trigonometric functions of  using the known values of sin  and cos . tan  sin  cos  == 4 5 3 5 – = 4 3 – cot  cos  sin  = = 4 5 3 5 – = 3 4 –

EXAMPLE 1 Find trigonometric values csc   sin  == 1 4 5 = 5 4 sec   cos  = = 3 5 – 1 = 5 3 –

EXAMPLE 2 Simplify a trigonometric expression Simplify the expression tan ( –  ) sin . π 2 Cofunction Identity tan ( –  ) sin  π 2 cot  sin  = Cotangent Identity = ( ) ( sin  ) cos  sin  Simplify. = cos 

EXAMPLE 3 Simplify a trigonometric expression 2 Simplify the expression csc  cot  +.  sin  Reciprocal Identity 2 csc  cot  +  sin  csc  cot  + csc  2 = Pythagorean Identity = csc  (csc  – 1) + csc  2 Distributive property = csc  – csc  + csc  3 Simplify. = csc  3

GUIDED PRACTICE for Examples 1, 2, and 3 Find the values of the other five trigonometric functions of . 1 6 1. cos , 0 <  < = π 2 SOLUTION sin  = 35 6 sec  = 6 csc  cot  = 6 35 =

GUIDED PRACTICE for Examples 1, 2, and 3 Find the values of the other five trigonometric functions of . 2. sin  =, π <  3π3π 2 –3 7 SOLUTION cos  – = 2 10 7 tan  = 20 3 10 csc  = 7 3 – sec  = – 7 20 10 cot  2 10 3 = –

GUIDED PRACTICE for Examples 1, 2, and 3 3. sin x cot x sec x Simplify the expression. 1 ANSWER 4. tan x csc x sec x 1 ANSWER cos –1 π 2 –  1 + sin (–  ) 5. –1 ANSWER

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