sin 2θ when cos θ = 7/25, θ in quadrant IV Step by Step

Step by step

The question

Question: find sin 2θ when cos θ = 7/25, θ in quadrant IV.

We know cos θ, and the quadrant.

sin 2θ =

cos θ = 725, quadrant IV

1 Draw the triangle

Take a point (x, y) at distance r from O.

cos θ is x over r.

So x = 7 and r = 25.

cos θ = xr = 725

We draw it in quadrant IV: a right triangle from O to the point.

2 Find the third side

Pythagoras: x² + y² = r².

y² + 7² = 25²

7 squared is 49.

25 squared is 625.

y² + 49 = 625

Take away 49 on both sides: y² = 576.

y² = 625 − 49 = 576

The square root of 576 is 24.

Plus or minus.

y = ±√576 = ±24

In quadrant IV, y is negative: below the x-axis.

So y = −24.

y = −24

3 Find sin θ

Sin θ is y/r: −24/25.

So sin θ = −24/25.

sin θ = yr = −2425 = −2425

4 Write the rule

The double-angle rule: sin 2θ = 2 sin θ cos θ.

sin 2θ = 2 sin θ cos θ

Careful!

sin 2θ is not 2 times sin θ.

The 2 is inside: it doubles the angle, not the answer.

−48/25 is not between −1 and 1, and a sine always is.

Careful: not sin 2θ = 2 sin θ = −4825 ✗

5 Work it out

Put in sin θ = −24/25 and cos θ = 7/25.

sin 2θ = 2 × (−2425) × 725

Tops: 2 times 24 times 7 is 336.

Bottoms: 25 times 25 is 625.

One minus sign: the answer is negative.

2 × 24 × 7 = 336, 25 × 25 = 625

So sin 2θ = −336/625.

sin 2θ = −336625

6 Check

Let's check with the calculator.

The reference angle is tan⁻¹ of 24/7: about 73.74°.

In quadrant IV, θ is about 286.26°.

θ ≈ 360° − 73.74° = 286.26°

Twice that is 572.52°.

sin of it is about −0.538.

sin 2θ ≈ sin 572.52° ≈ −0.538

And −336/625 is −0.538.

The same!

So it is right.

−336625 = −0.538

Let's look back.

First: draw the triangle.

Then: find the third side.

Then: find sin θ.

Then: write the rule.

Then: work it out.

Last: check.

Answer