If tan ⁡x=60/11, and π<x<3π/2, what is sin⁡(x−π)?

Question

What is the correct way to apply the Difference Identity to find sin75∘?

a) sin75∘=sin(120∘−45∘)=sin(120∘)cos(45∘)+cos(120∘)sin(45∘)

b) sin75∘=sin(120∘−45∘)=sin(120∘)sin(45∘)−cos(120∘)cos(45∘)

c) sin75∘=sin(120∘−45∘)=sin(120∘)cos(45∘)−cos(120∘)sin(45∘)

d) sin75∘=sin(120∘−45∘)=sin(120∘)sin(45∘)+cos(120∘)cos(45∘)

2) If tan ⁡x=60/11, and π<x<3π/2, what is sin⁡(x−π)?

Enter your answer as a fraction in simplest form, like this: 3/14

3) If cos x = 8/17 and 3π/2 <x <2π, what is sin (x - π)?

Type your answer as a fraction in the simplest way, like this: 3/14

4) If 180°<θ<270°, and sin⁡θ=−3/4, what is the value of sin(−θ)?

Enter your answer as a fraction in simplest form, like this: 3/14

Answer 1

1) What is the correct way to apply the Difference Identity to find sin75∘?

a) sin75∘=sin(120∘−45∘)=sin(120∘)cos(45∘)+cos(120∘)sin(45∘)

b) sin75∘=sin(120∘−45∘)=sin(120∘)sin(45∘)−cos(120∘)cos(45∘)

c) sin75∘=sin(120∘−45∘)=sin(120∘)cos(45∘)−cos(120∘)sin(45∘)

d) sin75∘=sin(120∘−45∘)=sin(120∘)sin(45∘)+cos(120∘)cos(45∘)

The difference of sines identity

Sin(A - B) = sinA cosB - cosA sinB

The answer for #3 is c.

Answer choice "a" can be eliminated because the Difference Identity for Sines has a "subtraction sign'' in the identity,

Choice "b" can be eliminated because in the Difference Identity for Sines sin and cos of A and B alternate on either side of the operator.

Choice "d" can be eliminated because in the Difference Identity for Sines sin and cos of A and B alternate on either side of the operator and the operator should be subtraction not addition.

2) If tan ⁡x=60/11, and π<x<3π/2, what is sin⁡(x−π)?

Enter your answer as a fraction in simplest form, like this: 3/14

tan x = y/x so y = -60 and x = -11, so we use the Pythagorean Theorem to find r (the hypotenuse)

(-60)² + (-11)² = r²

3600 + 121 = r²

(3721)^.5 = (r²)^.5

r = 61

Therefore sin x = -60/61and cos x = -11/61

sin⁡(x−π) = sin x cos ⁡π - cosx sin ⁡π

sin⁡(x−π) = (-60/61)(-1) - (-11/61)(0)

sin⁡(x−π) = 60/61

3) If cos x = 8/17 and 3π/2 <x <2π, what is sin (x - π)?

Type your answer as a fraction in the simplest way, like this: 3/14

cos x = x/r so x = 8 and r = 17, so we use the Pythagorean Theorem to find y (the sine)

(17)² - (8)² = y²

225 = y²

(225)^1/2 = (y²)^1/2

y = 15

Since theta lieu in the 4th quadrant, y is negative.

Therefore x = 8, y = -15 and r = 17

Therefore sin x = -15/17 and cos x = 8/17

sin⁡(x−π) = sin x cos ⁡π - cosx sin ⁡π

sin⁡(x−π) = (-15/17)(-1) - (8/17)(0)

sin⁡(x−π) = 15/17