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More Trigonometric Identities Worksheet
Concepts:
• The Addition and Subtraction Identities for Sine and Cosine
• The Cofunction Identities for Sine and Cosine
• Identities That You Should Learn
(Section 7.3)
1. Evaluate the six trigonometric functions at θ =
calculator.
π
.
12
Do not use your
2. Simplify cos(π + x).
3. Prove the identity tan
π
2
− x = cot(x).
4. (Exercise 47 in Section 7.2 of your textbook)
The terminal ray of x is in the first quadrant, the terminal ray of y is
12
in the second quadrant, sin(x) = 45 , and cos(y) = − 13
.
(a) Find the exact values of sin(x + y), cos(x + y), and tan(x + y).
(b) In which quadrant is the terminal ray of the angle x + y found?
(Assume all angles are in standard position.)
π
π
.
5. Find the exact value of sin
−π+
4
6
6. Express cos(α + β + γ) in terms of sines and cosines of α, β, and γ.
7. Prove that the difference quotient for g(x) = cos(x) is equal to
cos(h) − 1
sin(h)
cos(x)
− sin(x)
.
h
h
8. Express sin2 (3x) in terms of constants and first powers of the cosine
function.
1
9. Express cos4 (5x) in terms of constants and first powers of the cosine
function.
2 x
10. Express sin
in terms of constants and first powers of the cosine
2
function.
2
11. (Exercise 9 from section 7.3 of your textbook)
A rectangle is inscribed in a semicircle of radius 3 inches and the radius
to the corner makes an angle of t radians with the horizontal, as shown
in the picture.
• (x, y)
t
(a) Express the horizontal length, vertical height, and area of the
rectangle in terms of x and y.
(b) Express x and y in terms of sine and cosine (NOTE: This is not
a unit circle.)
(c) Use parts (a) and (b) and suitable identities to show that the area
of the rectangle is given by
A = 9 sin(2t).
12. (Exercise 90 from section 7.3 of your textbook)
To avoid a steep hill, a road is being built in straight segments from P
to Q and from Q to R; it makes a turn of t radians at Q as shown in
the figure. The distance from P to S is 40 miles, and the distance from
R to S is 10 miles. Use suitable trigonometric functions to express:
(a) c in terms of b and t. [Hint: Place the figure on a coordinate plane
with P and Q on the x-axis, with Q at the origin. Then what are
the coordinates of R?]
(b) b in terms of t
(c) a in terms of t [Hint: a = 40 − c; use parts (a) and (b).]
(d) Use parts (b) and (c) and a suitable identity to show that the
length a + b of the road is
t
40 + 10 tan
.
2
3
R
STEEP HILL
b
10
t
P
a
Q
40
4
c
S