Right triangle trigonometry: SOHCAHTOA explained

Q: Why is my calculator giving the wrong trig answer?

It is almost always in radian mode. In degree mode \sin 30 = 0.5. In radian mode the calculator reads 30 as 30 radians and shows about -0.988. Switch to degrees and test \sin 30 again.

Q: When do I use inverse trig functions?

Use them when you know two sides and want an angle. \sin^{-1}(0.7) means the angle whose sine is 0.7. It does not mean 1 divided by \sin 0.7. On most calculators it is the second function of the sin key.

The key idea

Pick one acute angle and imagine standing at it. The side across from you is the opposite side. The leg touching you is the adjacent side. The hypotenuse is across from the right angle and never changes.

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

SOHCAHTOA is the memory trick: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. These ratios depend only on the angle, not on how big the triangle is, because right triangles with the same angles are similar.

Which tool to use
You know	You want	Use
An acute angle and one side	Another side	sin, cos or tan, then solve the equation
Two sides	An angle	$\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$
Two sides	The third side	The Pythagorean theorem

Sine and cosine swap

The two acute angles of a right triangle add to 90°. The side opposite one of them is adjacent to the other, so the sine of one angle is the cosine of the other.

\sin x^\circ = \cos(90^\circ - x^\circ)

Worked examples

Example 1: write the three ratios

Problem Right triangle ABC has its right angle at C. The legs are $BC = 8$ and $AC = 15$ , and the hypotenuse is $AB = 17$ . Find $\sin A$ , $\cos A$ and $\tan A$ .

Stand at angle A. The side across from A is BC = 8, so it is opposite. The leg touching A is AC = 15, so it is adjacent. The hypotenuse is 17.
Sine is opposite over hypotenuse.
$\sin A = \frac{8}{17}$
Cosine is adjacent over hypotenuse.
$\cos A = \frac{15}{17}$
Tangent is opposite over adjacent.
$\tan A = \frac{8}{15}$

Answer $\sin A = \frac{8}{17}$ , $\cos A = \frac{15}{17}$ , $\tan A = \frac{8}{15}$

Example 2: angle of elevation

Problem You stand 40 meters from the base of a building. The angle of elevation from the ground where you stand to the top of the building is 52°. How tall is the building, to the nearest tenth of a meter?

Draw it. The building and the ground meet at a right angle. The 40 meters along the ground is adjacent to the 52° angle, and the height h is opposite it.
Opposite and adjacent means tangent.
$\tan 52^\circ = \frac{h}{40}$
Multiply both sides by 40.
$h = 40 \tan 52^\circ$
In degree mode, $\tan 52^\circ \approx 1.2799$ , so $h \approx 40(1.2799) \approx 51.2$ .

Answer About 51.2 meters.

Example 3: find an angle with inverse trig

Problem A wheelchair ramp rises 3 feet over a horizontal distance of 20 feet. What angle does the ramp make with the ground, to the nearest tenth of a degree?

The rise of 3 feet is opposite the angle, and the 20 feet along the ground is adjacent to it.
Opposite and adjacent means tangent.
$\tan\theta = \frac{3}{20}$
To get the angle itself, use inverse tangent.
$\theta = \tan^{-1}\left(\frac{3}{20}\right)$
In degree mode, $\tan^{-1}(0.15) \approx 8.53$ , which rounds to 8.5.

Answer About $8.5^\circ$ .

Example 4 (test-hard): sine equals cosine

Problem If $\sin(2x + 10)^\circ = \cos(3x - 5)^\circ$ and both angles are acute, what is x?

For acute angles, the sine of one angle equals the cosine of another exactly when the two angles add to 90°.
Add the angles and set the sum equal to 90.
$(2x + 10) + (3x - 5) = 90$
Combine like terms.
$5x + 5 = 90$
Subtract 5.
$5x = 85$
Divide by 5.
$x = 17$
Check: the angles are 44° and 46°. Both are acute and they add to 90°.

Answer $x = 17$

Common mistakes

Radian mode. $\sin 30$ should be 0.5. If it shows about $-0.988$ , switch the calculator to degrees.
Mixing up opposite and adjacent. They depend on which angle you stand at. Fix: put your finger on the angle first, then name the sides.
Calling the hypotenuse adjacent. The hypotenuse touches the angle too, but adjacent always means the leg, never the hypotenuse.
Reading $\sin^{-1}$ as 1 over sine. $\sin^{-1}(0.7)$ means the angle whose sine is 0.7. It is not $\frac{1}{\sin 0.7}$ .
Rounding too early. Keep the full calculator value until the last step, then round once.

Quick methods

Practice

5 practice questions

Right triangle PQR has its right angle at R. $PR = 5$ , $QR = 12$ and $PQ = 13$ . What is $\cos P$ ?
1. $\frac{5}{13}$
2. $\frac{12}{13}$
3. $\frac{5}{12}$
4. $\frac{12}{5}$
Show answer
Answer: $\frac{5}{13}$
Standing at P, the adjacent leg is PR = 5 and the hypotenuse is 13, so $\cos P = \frac{5}{13}$ . $\frac{12}{13}$ is $\sin P$ , $\frac{12}{5}$ is $\tan P$ , and $\frac{5}{12}$ is $\tan Q$ .
A kite string is 60 feet long, pulled straight, and makes a 40° angle of elevation with the ground. About how high is the kite above the point where the string is held?
1. About 38.6 feet
2. About 46.0 feet
3. About 50.3 feet
4. About 71.5 feet
Show answer
Answer: About 38.6 feet
The string is the hypotenuse and the height is opposite the 40° angle, so the height is $60 \sin 40^\circ \approx 38.6$ . 46.0 uses cosine, 50.3 uses tangent, and 71.5 divides 60 by $\tan 40^\circ$ .
In a right triangle, the side opposite angle A is 7 and the hypotenuse is 10. What is angle A, to the nearest degree?
1. $35^\circ$
2. $44^\circ$
3. $46^\circ$
4. $55^\circ$
Show answer
Answer: $44^\circ$
Opposite over hypotenuse is sine, so $A = \sin^{-1}(0.7) \approx 44.4$ , which rounds to 44°. 46° comes from $\cos^{-1}(0.7)$ , 35° from $\tan^{-1}(0.7)$ , and 55° from $\tan^{-1}\left(\frac{10}{7}\right)$ .
Which expression is equal to $\sin 28^\circ$ ?
1. $\cos 28^\circ$
2. $\cos 62^\circ$
3. $\sin 62^\circ$
4. $\tan 62^\circ$
Show answer
Answer: $\cos 62^\circ$
$28 + 62 = 90$ , so $\sin 28^\circ = \cos 62^\circ$ , both about 0.469. $\cos 28^\circ$ and $\sin 62^\circ$ equal each other (about 0.883), not $\sin 28^\circ$ . $\tan 62^\circ$ is about 1.88.
In right triangle ABC, the right angle is at C and $\sin A = \frac{3}{5}$ . What is $\cos B$ ?
Show answer
Answer: $\frac{3}{5}$
The side opposite A is the side adjacent to B, and both angles share the same hypotenuse. So $\cos B = \sin A = \frac{3}{5}$ , which is 0.6.

Frequently asked questions

What does SOHCAHTOA stand for?

It packs the three definitions into one word. Sine is opposite over hypotenuse (SOH), cosine is adjacent over hypotenuse (CAH), and tangent is opposite over adjacent (TOA). Opposite and adjacent always mean relative to the angle you are working with.

Why is my calculator giving the wrong trig answer?

It is almost always in radian mode. In degree mode $\sin 30 = 0.5$ . In radian mode the calculator reads 30 as 30 radians and shows about $-0.988$ . Switch to degrees and test $\sin 30$ again.

When do I use inverse trig functions?

Use them when you know two sides and want an angle. $\sin^{-1}(0.7)$ means the angle whose sine is 0.7. It does not mean 1 divided by $\sin 0.7$ . On most calculators it is the second function of the sin key.

Why is sin x equal to cos(90° - x)?

The two acute angles in a right triangle add to 90°. The side opposite one angle is the side adjacent to the other, and the hypotenuse is shared. So the sine of one angle and the cosine of the other are the very same fraction.

Right triangle trigonometry and SOHCAHTOA

The key idea

Sine and cosine swap

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Try asking Ducky

The key idea

Sine and cosine swap

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Related

Try asking Ducky