Alliance Form 3 P1 Q8 — Trigonometry: Complementary Angles

KCSE (Form 3) Form 3 Trigonometry

Published

The Question

“Given that the sine of (x + 60) degrees is equal to the cosine of 2x degrees, find the tangent of (x + 60) degrees, correct to four significant figures.”

1

Recall the complementary angle rule

Sine and cosine are complementary functions, which means the sine of an angle equals the cosine of ninety minus that same angle. So whenever the sine of one angle equals the cosine of another, those two angles must add up to ninety degrees. This is the single idea that unlocks the whole question.

sin⁡A=cos⁡(90∘−A)\sin A = \cos\left(90^\circ - A\right)
sin⁡A=cos⁡B  ⇒  A+B=90∘\sin A = \cos B \;\Rightarrow\; A + B = 90^\circ
2

Set up the equation from the two angles

Here the first angle is x plus sixty and the second angle is two x. Because their sines and cosines match, the rule tells us they must add up to ninety degrees. Write that sum as an equation.

(x+60)+2x=90∘\left(x + 60\right) + 2x = 90^\circ
3

Collect terms and solve for x

Combine the x terms on the left, then subtract sixty from both sides to isolate the x term, and finally divide by three. This gives the value of x.

3x+60=903x + 60 = 90
3x=303x = 30
x=10∘x = 10^\circ
4

Answer what was actually asked

The question does not want x on its own. It wants the tangent of x plus sixty. Substitute x equals ten to get the angle inside the tangent, then read off its tangent from tables or a calculator to four significant figures.

x+60=10+60=70∘x + 60 = 10 + 60 = 70^\circ
tan⁡70∘=2.748\tan 70^\circ = 2.748

Final Result

The tangent of (x + 60) degrees is 2.748, correct to four significant figures.

Why this method works

The method works because sine and cosine describe the two acute angles of the same right-angled triangle, which always sum to ninety degrees. So the sine of one acute angle is identical to the cosine of the other. When we are told the sine of one angle equals the cosine of another, the cleanest reading is that those angles are the complementary pair, so they must add to ninety. That turns a trigonometric statement into a simple linear equation in x, no sine or cosine values needed. Once x is found, the tangent is just a direct lookup.

✓

With x = 10, sin(70) = cos(20) = 0.9397, confirming sin(x+60) = cos 2x holds, and tan 70 = 2.748.