10.3 Verify Trigonometric Identities

1. 10.3 Verify Trigonometric Identities

3. Trigonometric Identities • Trigonometric Identity: a trigonometric equation that is true for all values of the variable for which both sides of the equation are defined. • There are 5 fundamental Trigonometric Identities. (See page 628 in your book.)

5. 1+ tan 2θ = sec2θ

6. 1+ cot2θ= csc2θ

7. Section 9.3 Page 572

8. Given that sin q = and < q <π, find the values of the other five trigonometric functions of q . 4 4 4 1 = 5 5 5 4 3 3 4 π 2 2 2 = cosq 1 – ( ) 2 5 5 5 5 Subtract ( ) from each side. 2 Substitute for sin q. 2 ( ) + cosq 2 cosq = + cosq = – 9 25 – = cosq 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.

9. 1 1 csc q = = = sin q 1 1 = sec q = = cos q cos q sin q – sin q cos q 5 5 4 3 3 4 3 4 3 4 – – – tan q = = = 3 5 3 5 5 5 5 5 4 4 – – cot q = = = STEP 2 Find the values of the other four trigonometric functions of q using the known values of sinq and cos q.

10. , 0< q < 1.cosq = ( )2 ( )2 2 2 1 1 sinq+cos q = = sinq = sinq = sin2q= 1 – 1 1 6 6 1 1 35 π 1 35 35 2 6 6 36 6 6 2 6 Subtract ( ) from each side. Substitute for cos q . 2 sin q+ sin2q = Find the values of the other five trigonometric functions of q. SOLUTION STEP 1 Find sin q. Write Pythagorean identity. Simplify. Take square roots of each side. Because q is in Quadrant I, sinq is positive.

11. 1 1 6 csc q = = = = = sin q 35 35 35 35 1 = cos q sin q cos q sin q cos q 35 35 1 35 1 1 1 6 = cot q = = 6 35 6 6 35 6 6 6 = 1 tan q sec q = 35 = = STEP 2 = 6

12. + cosq = – – = cosq 2.sin q = , π < q < 2 2 1 sinq+cos q = – 3 – 3 – 3 7 7 – 3 7 1 – ( )2 –3 2 10 2 10 3π 7 2 7 cos q 7 7 2 2 cosq Substitute for sin q . ( )2+ cos2q= 1 40 = Subtract from each side. 49 = Find the values of the other five trigonometric functions of q. SOLUTION STEP 1 Find cos q. Write Pythagorean identity. Simplify. Take square roots of each side. Because q is in Quadrant lII, cosq18 is negative.

13. 1 1 = = = sin q cot q = 1 = cos q cos q sin q 2 10 7 3 2 10 3 – 2 10 – csc q 3 7 3 7 7 – 7 = – = 7 10 1 – sec q = = 20 –

14. 10.3 Assignment Day 1 Page 631, 3-9 all

15. 10.3 Verify Trigonometric Identities, day 2

16. 1 2 Simplify the expression csc q cot q+ . sin q 2 csc qcot q+csc q = 1 2 2 csc q cot q+ = csc q(csc q – 1)+ csc q sin q 3 = csc q – cscq + cscq 3 = csc q Reciprocal identity Pythagorean identity Distributive property Simplify.

17. Simplify the expression tan ( – q ) sin q. cot qsin q tan ( – q ) sin q = = ( ) ( sin q ) cos q sin q π π 2 2 Cofunction identity Cotangent identity = cos q Simplify.

18. cos x 1 sin x ANSWER · · 1 cos x sin x Simplify the expression. 3.sin x cot x sec x Substitute identity functions Simplify

19. · sin x 1 1 cos x sin x cos x tan x csc x 4. sec x ANSWER 1 Simplify the expression. Substitute identity functions Simplify

20. π 2 sin ( θ) – 1 sin ( θ) – 1 ANSWER – 1 – 1(sin ( θ) – 1) 1 – sin ( θ) – q cos –1 5. 1 + sin (– q) Simplify the expression. Substitute Cofunction identity; Substitute Negative Angle identity Factor Simplify

21. 10.3 Assignment Day 2 Page 631, 10-19 all

22. 10.3 Verify Trigonometric Identities, day 3

24. Verifying Trigonometric Identities • When verifying an identity, begin with the expression on one side. • Use algebra and trigonometric properties to manipulate the expression until it is identical to the other side.

25. 2 Verify the identity = sin q. 1 – = 2 2 2 2 2 2 2 sec q–1 sec q sec q sec q–1 sec q sec q sec q 2 1 = 1 –( ) sec q 2 = 1 – cosq 2 = sinq Write as separate fractions. Simplify. Reciprocal identity Pythagorean identity

26. cosx cos x Verify the identity secx + tanx = . = 1 – sin x 1 – sin x = 1 sin x 1 + tan x = + cos x cos x cos x 1 + sin x 1 + sin x = = cos x cos x 1 – sin x 1 – sin x Multiply by 1 – sin x 1 – sin x 2 1 – sin x = cos x (1 – sin x) 2 cos x = cos x (1 – sin x) secx+ tanx Reciprocal identity Tangent identity Add fractions. Simplify numerator. Pythagorean identity Simplify.

27. Shadow Length A vertical gnomon (the part of a sundial that projects a shadow) has heighth. The length sof the shadow cast by the gnomon when the angle of the sun above the horizon is q can be modeled by the equation below. Show that the equation is equivalent to s = hcotq . h sin (90° – q ) = sinq s

28. = h sin ( – q ) π = 2 sinq = h cos q sin q h sin (90° – q ) sinq SOLUTION Simplify the equation. s Write original equation. Convert 90° to radians. Cofunction identity = h cotq Cotangent identity

29. cot (– q ) = 1 1 = tan (– θ) –tan ( θ) Verify the identity. 6.cot (– q ) = – cotq SOLUTION Reciprocal identity Negative angle identity Reciprocal identity = – cotθ

30. = cos2x 1 sin2 x Verify the identity. 7.csc2x (1 – sin2x) = cot2x SOLUTION csc2x (1 – sin2x ) = csc2xcos2x Pythagorean identity Reciprocal identity = cot2 x Tangent identity and cotangent identities

31. cosx csc x tanx 1 sinx sin x cosx sinx cosx = cosx csc x = cos x Verify the identity. 8.cosx csc x tanx = 1 SOLUTION Tangent identity and cotangent identities Reciprocal identity = 1 Simplify

32. (tan2x + 1)(cos2x – 1) 1 cos2x (–sin2x) = – sec2x (–sin2x) = Verify the Identity. 9. (tan2x + 1)(cos2x – 1) = – tan2x SOLUTION Pythagorean identity Reciprocal identity = –tan2x Tangent identity and cotangent identities

34. 10.3 Assignment, day 3 Page 631, 25-32 all