Given sin(theta)=1/4 find the exact values of sin(2theta),cos(2theta), and tan(2theta)

Answer 1

√Hi Kayla H.,

We can use the double-angle formulas for the functions of 2θ, but we will still need to know the θ's.

For right triangles: sinθ = opp/hyp, cosθ = adj/hyp, and tanθ = opp/hyp.

With sinθ = 1/4, we know opp = 1, hyp = 4, and adj = unknown.

We can use a² + b² = c² which rearranges to a = ± [ c² - b² ]^1/2 to find our unknown (well let a =adj).

a = ± [ 4² -1² ]^1/2 = ± √15 = adj, (choosing the positive for a dimension)

We now now all sides of the triangle. The functions of θ are now:

sinθ = 1 / 4, cosθ = √15 / 4, tanθ = 1 / √15 = √15 / 15

Now we can use the double-angle formulas:

sin2θ = 2*sinθ*cosθ = 2*(1/4)*(√15/4) = √15 / 8;

cos2θ = cos²θ - sin²θ = (√15/4)² - (1/4)² = 15/16 - 1/16 = 14/16 = 7/8;

tan2θ = 2tanθ / (1 - tan²θ) = 2*(√15/15) / [ 1 - (√15/15)² ] = √15 / 7.

I hope this helps, Joe.

Answer 2

sinθ = 1/4

If we assume that θ is acute, then since cos²θ + sin²θ = 1, we get cosθ = √15/4

So, sin(2θ) = 2sinθcosθ = 2(1/4)(√15/4) = √15/8

cos(2θ) = 1 - 2sin²θ = 1 - 2(1/4)² = 7/8

tan(2θ) = sin(2θ) / cos(2θ) = √15/7

Answer 3

First, you need to evaluate sin θ = 1/4.

To do this, you need to apply the inverse trig function for sine to both sides:

sin^-1 (sin θ) = sin^-1 (1/4)

The equation simplifies down to:

θ = sin^-1 (1/4) = sin^-1 (0.25) = 0.25268

Now, apply your value for across the sin2θ, cos 2θ, and tan 2θ.

sin2θ = sin (2⋅0.25268) = sin (0.50536) = 0.00882

cos 2θ = cos (2⋅0.25268) = cos (0.50536) = 0.99996

tan 2θ = tan (2⋅0.25268) = tan (0.50536) = 0.00882

Answer 4

sin(theta)=14

step 1) Take the inverse of both sides

sin^-1(sin(theta))=sin^-1(1/4)

this results in the following equation: theta=sin^-1(1/4)

step 2) When you put sin^-1(1/4) into the calculator you get ~14.478

step 3) use the theta to plug into the equaions

sin(2*14.478) = .484

cos(2*14.478) = .875

tan((2*14.478) = .258