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Trigonometric Values of q with Given sinq and cosq

Calculate the values of the other five trigonometric functions (tanq, cotq, cscq, secq) using the given values of sinq and cosq.

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Trigonometric Values of q with Given sinq and cosq

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

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

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

  4. 1 1 csc q = = = sin q 1 1 = sec q = = cos q – 3 4 5 5 – 4 3 5 5 EXAMPLE 1 Find trigonometric values

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

  6. 1 2 Simplify the expression csc q cot q+ . sin q 2 csc qcotq+csc q = 1 2 2 csc q cot q+ = csc q(csc q – 1)+ csc q sin q 3 = cscq – cscq + cscq 3 = cscq EXAMPLE 3 Simplify a trigonometric expression Reciprocal Identity Pythagorean Identity Distributive property Simplify.

  7. , 0< q < 1.cosq = sinq = 35 1 π 6 2 6 6 35 csc q = 35 sec q = 6 35 cot q = 35 for Examples 1, 2, and 3 GUIDED PRACTICE Find the values of the other five trigonometric functions of q. SOLUTION

  8. = – = cosq 2.sin q = , π < q < –3 2 10 3π 7 2 10 7 3 7 3 2 7 10 10 3 – 20 20 for Examples 1, 2, and 3 GUIDED PRACTICE Find the values of the other five trigonometric functions of q. SOLUTION – csc q cot q = – tan q = sec q =

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

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