Download trigidentitiesworksheet

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
15
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