PRECAL &TRIG QUESTIONS DUE AT 10:30
BrindiaJ0124
precal2.pdf
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
Please show ALL work and pertinent calculations on these examination papers. Place your answers in the appropriate answer blank, when provided. You may find the following expressions useful.
1. tan(α + β) = tan(α) + tan(β)
1 − tan(α) tan(β)
2. sin (
x 2
)
= ±
√
1 − cos(x)
2 cos
(
x 2
)
= ±
√
1 + cos(x)
2
3. tan (
x 2
)
= ±
√
1 − cos(x)
1 + cos(x) =
sin(x)
1 + cos(x) =
1 − cos(x)
sin(x)
4. sin2(x) = 1 − cos(2x)
2 cos2(x) =
1 + cos(2x)
2
5. tan2(x) = 1 − cos(2x)
1 + cos(2x)
6. sin(α) sin(β) = 1 2 [cos(α − β) − cos(α + β)]
7. cos(α) cos(β) = 1 2 [cos(α − β) + cos(α + β)]
8. sin(α) cos(β) = 1 2 [sin(α + β) + sin(α − β)]
9. cos(α) sin(β) = 1 2 [sin(α + β) − sin(α − β)]
10. sin(α) + sin(β) = 2 sin (
α+β
2
)
cos (
α−β
2
)
11. sin(α) − sin(β) = 2 sin (
α−β
2
)
cos (
α+β
2
)
12. cos(α) + cos(β) = 2 cos (
α+β
2
)
cos (
α−β
2
)
13. cos(α) − cos(β) = −2 sin (
α+β
2
)
sin (
α−β
2
)
–Created/Revised by Mr. Sever 10/18/20 Page 0 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
1. Verify the identity: sin2(x) + cos2(x) + tan2(x) = sec2(x)
2. Verify the identity: cos(2x)
tan2(x) − 1 = − cos2(x)
3. Verify the identity: cos2(x) − 1
sin(2x) = −
1
2 tan(x)
–Created/Revised by Mr. Sever 10/18/20 Page 1 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
4. Without the aid of a calculator, find the exact value of sin(105◦).
5. Given that sin(θ) = 12 13
and that θ is in Quadrant II with π 2
< θ < 3π 4
, find cos(2θ).
6. Use the appropriate sum-to-product formula (See cover page.) to find the exact value of
cos(15◦) + cos(75◦).
–Created/Revised by Mr. Sever 10/18/20 Page 2 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
7. Solve (i.e., find the exact solutions) cos(2θ) = −
√ 3
2 on [0, 2π).
8. Solve (i.e., find the exact solutions) sin(2θ) = − cos(θ) on [0, 2π).
9. Solve (i.e., find the exact solutions) 1 − sin(θ) = 2 cos2(θ).
–Created/Revised by Mr. Sever 10/18/20 Page 3 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
10. Use the appropriate angle addition formula to solve cos(θ) cos(2θ) − sin(θ) sin(2θ) = 0 on [0, 2π).
11. State the domain of f (x) = 9x
16 − x2 .
12. Given f (x) =
−x + 6 , if x < −2 x2 , if − 2 ≤ x ≤ 3 4 , if x > 3
,
a) find f (−1)
b) find f (4)
c) find f (−4)
–Created/Revised by Mr. Sever 10/18/20 Page 4 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
13. State the domain and range of the function whose graph is given below.
1
−1 1
2
−2 2
3 Domain =
Range =
14. Let g(x) = −2 · f(x − 3) + 5. Which of the following successive transformations is a description of g?
A. The graph of f shifted 3 units to the left, reflected about the y-axis, stretched vertically by a factor of 2, and then
shifted up 5 units.
B. The graph of f shifted 3 units to the right, reflected about the y-axis, stretched vertically by a factor of 2, and then
shifted up 5 units.
C. The graph of f shifted 3 units to the left, reflected about the x-axis, stretched vertically by a factor of 2, and then
shifted up 5 units.
D. The graph of f shifted 3 units to the right, reflected about the x-axis, stretched vertically by a factor of 2, and then
shifted up 5 units.
E. None of A. through D. is correct
15. For f (x) = 5x − 4 and g(x) = 9 − x2, find (f ◦ g)(−2).
16. For f (x) = 1
√ x
and g(x) = x2 − 9x,
a) find (f ◦ g)(x)
b) state the domain of f ◦ g.
–Created/Revised by Mr. Sever 10/18/20 Page 5 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
17. Suppose that the domain of g is [−1, 1] and the domain of f is [0, 4]. Further, suppose that
(f ◦ g)(x) = |x| for all x in the domain of g and
(g ◦ f )(x) = x for all x in the domain of f .
Are f and g inverses (of each other)? Explain why or why not, briefly.
18. Let g(x) = −2 · f(x − 3) + 5. Assuming that the domain of f is [−3, 2] and the range of f is [0, 4], find: a) Domain of g = Range of g =
19. Let f (x) = 2x − 5
x + 3 . Find f −1(x).
–Created/Revised by Mr. Sever 10/18/20 Page 6 of 7
Math 1505, Examination 2 Name: Due: October 19, 2020 @ 11:00 PM
20. Find the domain of f (x) = √
x + √
7 − x.
21. Bonus: Without the aid of a calculator, find the exact value of sin (
cos−1 (
1
2
)
+ tan−1(1) )
.
–Created/Revised by Mr. Sever 10/18/20 Page 7 of 7
