Algebra and trigonometry fundamentals

Algebra -- Trig --

Writer's Note: The symbol "n" should really be referred to as "pi" or "?" To prevent confusion within the problem. Also, there is a difference between "squared" and "square root." The case in these problems is to use the square root, so "sqrt (number)." I've only managed the calculations because I have presumed the indicated changes.

Using the periodic properties of trigonometric functions, find the exact value of the expression

cos-

cos (8?/5) = cos (2? + (2?/5)) = cos (2?/5) = cos (72) = 0.31

cos (8?/5) = 0.31.

The point P. On the unit circle that corresponds to a real number t is:

{ 5-2 6 squared}

} Find csc (t)

P is on the unit circle, therefore the coordinates are (cos (t), sin (t)). This leads to the following calculations:

sin (t) = -2sqrt (6)/7, and csc (t) = 1/sin (t) = 1/(-2sqrt (6)/7) = 7/(-2sqrt (6)) = -7/2sqrt (6)

Answer: csc (t) = -7/2sqrt (6).

Use the reference to find the exact value of the expression

csc-

3

csc (4?/3) = csc (? + ?/3) -- csc (?/3) = -1/sin (?/3) = 1/sqrt (3/2) = -2/sqrt (3)

Answer: csc (4?/3) = -2/sqrt (3).

4. The point P. On the unit circle that corresponds to a real number t is:

{ 7 squared 3 }

{ - -, -- } Find the cot (t)

{ 4-4 }

P is on the unit circle, therefore coordinates are (cos (t), sin (t)). This leads to the following calculations:

cot (t) = cos (t)/sin (t) = x/y cos (t) = -sqrt (7)/4, sin (t) = -3/4

cot (t) = [-sqrt (7)/4] / (-3/4) = sqrt (7)/3

Answer: cot (t) = sqrt (7)/3.

5. What is the domain of the sine function?

Answer: The sine function is a continuous function along the x-axis, therefore the domain is all real numbers.

6. Find the reference angle for the following angle: -386degrees

Converting the degrees to a positive, one gets the following calculations:

360 -- 386 = -26-360 -- 26 = 334 degrees. The angle is in Quadrant IV. The reference angle can then be calculated as the following:

360 -- A = 360 -- 334 = 26 degrees.

Answer: The reference angle of -386 degrees is 26 degrees.

7. Use reference angles to find the exact value of the expression: -7n

tan-

4

tan (-7?/4) -7?/4 = -2? + ?/4

From the unit circle, one can see that the reference angle is ?/4

cos (?/4) = sqrt (2)/2

sin (?/4) = sqrt (2)/2

tan (?/4) = [sqrt (2)/2] / [sqrt (2)/2)] = 1

Answer: tan (-7?/4) = 1.

8. Using the unit circle, find the value of the trigonometric function: n tan-

6

From the unit circle, for ?/6, x = sqrt (3)/2 and y = 1/2

tan (?/6) = sin (?/6)/cos (?/6) = [sqrt (3)/2] / (1/2) = sqrt (3)

Answer: tan (?/6) = sqrt (3).

9. Find the value of sin (-120degrees)

Using the unit circle, one can find that the angle -120 degrees is found in the following calculations:

360 -- 120 = 240 degrees.

Looking at the unit circle, x = sin (240) = -1/2

Answer: sin (-120) = -1/2.

10. Find the reference angle for the following angle: 69 degrees

The angle lies on Quadrant I, therefore the reference angle is the same.

Answer: The reference angle is 69 degrees.

11. Using the unit circle, find the value of the trigonometric function: 3n

sec-

2

Using the unit circle, for 3?/2, x = 0, y = -1