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 2 θ = sec2θ

6. 1+ cot2 θ= csc2θ

7. Section 9.3
Page 572

8. Write Pythagorean identity.
Given that sin q = and < q < π, find the values of the other five trigonometric functions of q . 4 5 π 2
SOLUTION
STEP 1
Find cos q .
Write Pythagorean identity.
( ) + cos q 4 5 2 1 =
Substitute for sin q. 4 5
cos q 2 4 5
1 – ( ) =
Subtract ( ) from each side. 4 5 2
cos q 2 9 25 =
Simplify.
cosq 3 5 + – =
Take square roots of each side.
cosq 3 5 – =
Because q is in Quadrant II, cos q is negative.

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

10. Find the values of the other five trigonometric functions of q.1 6
1. cos q
, 0 < q < = π 2
SOLUTION
STEP 1
Find sin q .
sin q + cos q 2 = 1
Write Pythagorean identity.
sin q + 2 = 1
( )2 6
Substitute for cos q . 1 6
sin2 q = 1 –
( )2 1 6
Subtract ( ) from each side. 1 6 2
sin2 q = 35 36
Simplify.
sinq = 35 6
Take square roots of each side.
Because q is in Quadrant I, sin q is positive.
sinq = 35 6

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

12. Find the values of the other five trigonometric functions of q.
2. sin q = , π < q < 3π 2 –3 7
SOLUTION
STEP 1
Find cos q .
sin q + cos q 2 = 1
Write Pythagorean identity.
( )2 + cos2 q = 1
– 3 7
Substitute for sin q .
– 3 7
cos q 2 = 7
1 – ( )2
– 3
Subtract from each side.
– 3 7
cos q 2 40 49 =
Simplify.
cos q + – =
2 10 7
Take square roots of each side.
cosq – =
2 10 7
Because q is in Quadrant lII, cos q 18 is negative.

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

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

15. 10.3 Verify Trigonometric Identities, day 2

16. Simplify the expression csc q cot q + . 1 sin q2
Simplify the expression csc q cot q 1
sin q 2
csc q cot q + 1
sin q
csc q cot q + csc q 2 =
Reciprocal identity
= csc q (csc q – 1) + csc q 2
Pythagorean identity
= csc q – csc q + csc q 3
Distributive property
= csc q 3
Simplify.

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

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

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

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

21. 10.3 Assignment Day 2
Page 631, 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. Verify the identity = sin q . sec q – 1 sec q2
sec q
sec q – 1 2
sec q =
sec q 2 – 1
Write as separate fractions.
= 1 – ( ) 1
sec q 2
Simplify.
= 1 – cos q 2
Reciprocal identity
= sin q 2
Pythagorean identity

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

27. Shadow Length
A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of 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 = h cot q .
h sin (90° – q )
sin q = s

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

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

30. Verify the identity.
7. csc2 x (1 – sin2 x) = cot2 x
SOLUTION
csc2 x (1 – sin2 x ) = csc2 x cos2x
Pythagorean identity 1
sin2 x
= cos2 x
Reciprocal identity
= cot2 x
Tangent identity and cotangent identities

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

32. Verify the Identity.
9. (tan2 x + 1)(cos2 x – 1) = – tan2 x
SOLUTION
(tan2 x + 1)(cos2 x – 1)
= – sec2 x
(–sin2x)
Pythagorean identity 1
cos2 x
(–sin2x) =
Reciprocal identity
= –tan2 x
Tangent identity and cotangent identities

34. 10.3 Assignment, day 3
Page 631, all