Question

What is sin(4a) & cos(4a) if tan(a)=3

Answer 1

Answer:

• sin(4a) = -24/25
• cos(4a) = 7/25

Step-by-step explanation:

Your calculator can tell you these values:

  sin(4a) = sin(4·arctan(3)) = -0.96 = -24/25

  cos(4a) = cos(4·arctan(3)) = 0.28 = 7/25

_____

Some useful trig identities are ...

  sin(2a) = 2tan(a)/(1 +tan(a)^2)

  cos(2a) = (1 -tan(a)^2)/(1 +tan(a)^2)

Filling in the given value for tan(a), we find ...

  sin(2a) = 2(3)/(1+3^2) = 6/10 = 3/5

  cos(2a) = (1 -3^2)/(1 +3^2) = -8/10 = -4/5

Now, double-angle formulas are useful:

  sin(4a) = 2sin(2a)cos(2a) = 2(3/5)(-4/5) = -24/25

  cos(4a) = 1 -2sin(2a)^2 = 1 -2(3/5)^2 = 7/25

The desired trig function values are sin(4a) = -24/25; cos(4a) = 7/25.