Sine and Cosine of Complementary Angles

One leg faces one corner and lies beside the other.

The two acute angles add to 90°

The three angles of a triangle add up to 180°. In a right triangle one of them is 90°, so the other two add up to 180 − 90 = 90°. They are complementary angles.

So if one acute angle is x°, the other is (90 − x)°. In a right triangle with an angle of 37°, the other acute angle is 90 − 37 = 53°.

90°x°(90 − x)°

A right triangle whose acute angles are x° and (90 − x)°.

One side, seen from both angles

Give the triangle sides of 3, 4 and 5, with the hypotenuse 5 across from the right angle. The side of 3 is across from the angle x°, so it is the opposite side for x°.

Now stand at the other acute angle, (90 − x)°. The same side of 3 runs from that corner to the right angle, so for (90 − x)° it is the adjacent side. In the same way, the side of 4 is adjacent to x° and opposite (90 − x)°. The hypotenuse is the hypotenuse for both angles.

53490°x°(90 − x)°

The gold side of 3 is across from x° and beside (90 − x)°.

The same fraction twice

The sine of x° is the opposite side over the hypotenuse: sin x° = 3/5. The cosine of (90 − x)° is the adjacent side over the hypotenuse, and for that angle the adjacent side is the same side of 3, so cos(90 − x)° = 3/5 as well.

Both ratios divide the same 3 by the same 5, so they are the same number: sin x° = cos(90 − x)°. The side of 4 gives the other half: cos x° = 4/5 and sin(90 − x)° = 4/5, so cos x° = sin(90 − x)°.

Nothing in this used the numbers 3, 4 and 5, only which side touches which angle, so it is true for every acute angle. The sine of an angle equals the cosine of its complement, and the cosine of an angle equals the sine of its complement. This is called the cofunction relationship. It is where the name cosine comes from: the cosine of an angle is the sine of its complement.

The tangent follows the same picture, but it turns over rather than changing name: tan x° = 3/4 and tan(90 − x)° = 4/3.

Checked on the exact values

Take half an equilateral triangle with sides of 2: its sides are 1, √3 and 2, and its acute angles are 30° and 60°, which add to 90°. The side of 1 is across from the 30° angle and beside the 60° angle.

So sin 30° = 1/2 and cos 60° = 1/2: the same side over the same hypotenuse. The side of √3 gives sin 60° = √3/2 = cos 30°.

At 45° the angle is its own complement, since 45 + 45 = 90, so sin 45° = cos 45°. In the half square, both are 1/√2.

√32160°30°

Half an equilateral triangle. The gold side of 1 is across from 30° and beside 60°, so sin 30° = cos 60° = 1/2.

Using it

To write a sine as a cosine, keep the value and replace the angle by its complement. sin 40° = cos(90 − 40)° = cos 50°. A calculator agrees: sin 40° = 0.6428 and cos 50° = 0.6428, to four decimal places. In the same way, cos 25° = sin 65° and cos 12° = sin 78°.

It also solves equations. If sin x° = cos 50° and x is acute, then x is the complement of 50: x = 90 − 50 = 40.

The usual mistakes

Changing the name but not the angle. cos 40° is not sin 40°: at 40° the opposite and adjacent sides have different lengths. The angle has to change to its complement, 50°.

Using 180° instead of 90°. cos 140° takes 40 from 180, but the two acute angles of a right triangle share 90°, not 180°. 140° is not an angle of a right triangle at all.

A side from a ratio

The application below also finds sides. sin θ = opposite/hypotenuse, so multiplying both sides by the hypotenuse gives opposite = hypotenuse × sin θ. In the same way, adjacent = hypotenuse × cos θ. For a hypotenuse of 1.5 m and an angle of 14°, the opposite side is 1.5 sin 14° m.

Worked example: A Painter on a Ladder Who Needs the Angle at the Wall

Question A ladder 6 m long stands on level ground and leans against a vertical wall, making an angle of 76° with the ground. There is no calculator, only the values sin 76° = 0.9703 and cos 76° = 0.2419. (a) What angle does the ladder make with the wall, and what are the sine and cosine of that angle? (b) A painter stands on a rung 1.5 m from the top of the ladder, measured along the ladder. How far is that rung from the wall, and how far below the top of the ladder is it? Give both to 2 decimal places.

  1. 1.The wall meets the ground at a right angle, so the angles of the triangle at the foot of the ladder and at the top add to 90°. The angle between the ladder and the wall is 90 − 76 = 14°.

    wall76 deg14 degTR6 m1.5 mangle at the wall: 90 − 76 = 14 deg
    wall76 deg14 degTR6 m1.5 mangle at the wall: 90 − 76 = 14 deg
    The wall and the ground meet at a right angle, so the angle at the top is 90° − 76° = 14°.

  2. 2.The distance from the foot of the ladder to the wall is opposite the 14° angle and adjacent to the 76° angle, so sin 14° = cos 76°. The height up the wall is adjacent to 14° and opposite 76°, so cos 14° = sin 76°. (a) The ladder makes 14° with the wall, with sin 14° = 0.2419 and cos 14° = 0.9703.

    wall76 deg14 degTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703
    wall76 deg14 degTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703
    (a) The distance out from the wall is opposite 14° and adjacent to 76°: sin 14° = cos 76° = 0.2419, and cos 14° = sin 76° = 0.9703.

  3. 3.Let R be the rung, T the top of the ladder and K the point on the wall level with R. Triangle RKT is right-angled at K, its hypotenuse RT is 1.5 m, and its angle at T is the 14° between the ladder and the wall.

    wall76 deg14 degKTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at K
    wall76 deg14 degKTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at K
    Triangle RKT is right-angled at K, its hypotenuse RT is 1.5 m, and its angle at T is 14°.

  4. 4.RK is opposite the 14° angle, so RK = 1.5 sin 14° = 1.5 × 0.2419 = 0.36285 m. KT is adjacent to it, so KT = 1.5 cos 14° = 1.5 × 0.9703 = 1.45545 m.

    wall76 deg14 degKTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at KRK = 1.5 × sin 14 = 1.5 × 0.2419 = 0.36285KT = 1.5 × cos 14 = 1.5 × 0.9703 = 1.45545
    wall76 deg14 degKTR6 m1.5 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at KRK = 1.5 × sin 14 = 1.5 × 0.2419 = 0.36285KT = 1.5 × cos 14 = 1.5 × 0.9703 = 1.45545
    RK = 1.5 sin 14° = 1.5 × 0.2419 = 0.36285 m and KT = 1.5 cos 14° = 1.5 × 0.9703 = 1.45545 m.

  5. 5.(b) The rung is 0.36 m from the wall and 1.46 m below the top of the ladder. Check: 0.362852 + 1.455452 = 0.132 + 2.117 = 2.249, which is 1.52 = 2.25 to the accuracy of the values used.

    wall76 deg14 degKTR6 m1.5 m0.36 m1.46 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at KRK = 1.5 × sin 14 = 1.5 × 0.2419 = 0.36285KT = 1.5 × cos 14 = 1.5 × 0.9703 = 1.45545RK = 0.36 m, KT = 1.46 m
    wall76 deg14 degKTR6 m1.5 m0.36 m1.46 mangle at the wall: 90 − 76 = 14 degsin 14 = cos 76 = 0.2419cos 14 = sin 76 = 0.9703RT = 1.5 m, right angle at KRK = 1.5 × sin 14 = 1.5 × 0.2419 = 0.36285KT = 1.5 × cos 14 = 1.5 × 0.9703 = 1.45545RK = 0.36 m, KT = 1.46 m
    (b) The rung is 0.36 m from the wall and 1.46 m below the top of the ladder.

Answer: (a) 14°, with sin 14° = 0.2419 and cos 14° = 0.9703; (b) 0.36 m from the wall and 1.46 m below the top

Common mistakes

  • Using sin 76° for the distance from the wall. In the small triangle at the top, the side out from the wall is opposite the 14° angle, so it is 1.5 sin 14°; 1.5 sin 76° = 1.45545 m is the drop down the wall.
  • Deciding that a new angle needs a new value from a calculator. The two acute angles of a right-angled triangle share their sides, so sin 14° is already known: it is cos 76°.

More triangle trigonometry problems, worked step by step →

Practice Sine and Cosine of Complementary Angles in the app