Trigonometry

Question

1. find sin(5π/12) exactly

 

 

2. find sin π/8 exactly

 

3. given that  tan Θ= -3/4 and Θ is in quadrant II, find each of the following: cos 2Θ

Answer 1

1. Sin (5pi/12) = sin(pi/2 - pi/12) = sin(pi/2)cos(pi/12) - cos(pi/2)sin(pi/12) = cos(pi/12)

 

cos(pi/12) = cos[(pi/6)/2]

 

Now use half angle formula cos(x) = sqrt[ (1 + cos(2x))/2 ]

 

cos(pi/12) = sqrt[ (1 + cos(pi/6))/2 ]

 

pi/6 = 30 degrees so the cosine can be obtained from the 30-60-90 triangle.

 

 

2. Use the half angle formula sin(x) = sqrt[ (1 - cos(2x))/2 ]

 

sin(pi/8) = sin[(pi/4)/2] = sqrt[ (1 - cos(pi/4))/2 ]

 

pi/4 = 45 degrees so cos(pi/4) can be obtained from 45-45-90 triangle.

 

3. theta > 135 degrees since tan(theta) = - 3/4 and in second quadrant. So 2theta > 270 degrees, i.e. IV quadrant.

This means cos(2theta) is positive.

 

cos(2theta) = cos²(theta) - sin²(theta)

 

Since tan(theta) = - 3/4 (this is a 3-4-5 right triangle)  we have cos²(theta) = (4/5)² and sin²(theta) = (3/5)²