SOHCAHTOA (trigonometry): worked examples and method

Sine

Opposite 3 and hypotenuse 5. The sine is 3/5.

Cosine

Cosine uses the adjacent side. Adjacent 4 and hypotenuse 5. The cosine is 4/5.

Tangent

Tangent does not use the hypotenuse. Opposite 3 and adjacent 4. The tangent is 3/4.

Thirty degrees

Angle 30 degrees and hypotenuse 10. The opposite side is 5, because sin 30 = 1/2.

Forty-five

Angle 45 degrees and hypotenuse 10. sin 45 = √2/2, so the opposite side is 10 × √2/2 = 5√2. Both acute angles are 45 degrees, so the two shorter sides are equal.

Label first

Name the opposite, adjacent and hypotenuse for the angle you are using. Then choose the ratio.

Label from the angle you are using

Mark the right angle first. The side opposite it is the hypotenuse, and that label never moves. Then mark the acute angle named in the question. Look across the triangle from that angle: the side you see is the opposite. The remaining side, which touches the acute angle, is the adjacent. If the question uses the other acute angle, redraw the labels. Opposite and adjacent swap. The hypotenuse stays.

A ladder against a wall is a typical picture. The angle at the ground is the angle between the ladder and the floor. The wall height is opposite that angle. The distance along the ground is adjacent. The ladder itself is the hypotenuse. Labelling before you pick a ratio stops the common error of dividing the right pair of numbers with the wrong function.

Method: 1. Mark the right angle, then label the side across from it as the hypotenuse. 2. From the acute angle in the question, label the far side opposite and the touching side adjacent. 3. On a ladder, the wall is opposite the ground angle, so sin of that angle is wall divided by ladder.

Choose the ratio that fits the sides

Cover the side you do not have, in your head, and see which ratio is left. If you know the hypotenuse and want the opposite, sine links them. If you know the hypotenuse and want the adjacent, cosine links them. If you know the two legs, or you know one leg and want the other, tangent links them. You only need one ratio for each length.

Rearrange before you press the keys. If sin 40^° = fracopposite12, then the opposite is 12 × sin 40^°. If sin 35^° = frac4.2hypotenuse, then the hypotenuse is 4.2 ÷ sin 35^°. Multiplying when you should divide makes the hypotenuse shorter than a leg, which is impossible. The hypotenuse must be the longest side.

Method: 1. Use sin when you have the opposite and the hypotenuse. 2. Use cos when you have the adjacent and the hypotenuse. 3. If sin 40^°=dfracopposite12, multiply: the opposite is 12×sin 40^°.

Finding a side, and rounding at the end

Set the calculator to degrees. For an angle of 40^° and a hypotenuse of 12 cm, the opposite side is 12 × sin 40^°. The calculator shows about 7.713, which is 7.7 cm to 1 decimal place. Write more than one decimal in the working if you need the number again, and round only the final answer.

Two lengths can come from the same triangle. A height of 4.2 m opposite an angle of 35^° gives a hypotenuse of 4.2 ÷ sin 35^°, which is 7.3 m to 1 decimal place, and an adjacent side of 4.2 ÷ tan 35^°, which is 6.0 m to 1 decimal place. Check that 6.0 and 4.2 are both shorter than 7.3. They are, so the hypotenuse is still the longest side.

Method: 1. Multiply the hypotenuse by sine or cosine when that side is known. 2. Opposite 40^° in a hypotenuse of 12cm is 12×sin 40^°=7.7cm to 1 decimal place. 3. Check the hypotenuse stays longest: 7.3m is longer than both 6.0m and 4.2m.

Finding an angle

When two sides are known and the angle is not, write the ratio first, then use the inverse. Adjacent 9 cm and hypotenuse 15 cm means cos θ = 9/15 = 0.6. The angle is cos^-1(0.6), which is 53.1^° to 1 decimal place. The inverse of cosine is the right key because the ratio you wrote was cosine. Using sin^-1(0.6) answers a different question.

Give the angle in degrees and round at the end. 53.13^° to 1 decimal place is 53.1^°, not 53^°. If the question asks for the other acute angle as well, subtract from 90^°, because the two acute angles in a right-angled triangle add to 90^°. A quick sine or cosine check of your rounded angle should get you back close to the side you started with.

Method: 1. Write the ratio, then use the matching inverse. 2. Adjacent 9cm and hypotenuse 15cm give cosθ=9/15=0.6. 3. The angle is cos^-1(0.6)=53.1^° to 1 decimal place.

Angles of elevation, ladders and ramps

Real exam questions use trigonometry in a context: a ladder against a wall, a ramp, a kite on a string, the top of a tower seen from the ground. The angle of elevation is the angle measured upwards from the horizontal to your line of sight. The angle of depression is measured downwards from the horizontal.

Draw the right-angled triangle from the story before you do anything else. The ground is horizontal and a wall or tower is vertical, so there is a right angle at the base. Label the sides from the angle you are using: the hypotenuse, the opposite side and the adjacent side. Take care with the angle between the ladder and the wall, because the sides swap roles compared with the angle at the ground.

Method: 1. Sketch the right-angled triangle and mark the angle and the lengths you know. 2. Label the sides hypotenuse, opposite and adjacent from the marked angle. 3. Choose sine, cosine or tangent using the two sides involved. 4. Solve, round to the accuracy asked for and add the unit.

Tip: Angle of elevation is always measured from the horizontal, never from the vertical.

When the unknown is on the bottom of the fraction

Finding a hypotenuse or an adjacent side often needs a division. For example, if sin 15^° = 0.9/h, then h = frac0.9sin 15^°. Rearrange before you type, because multiplying by sin 15^° gives a hypotenuse that is shorter than the opposite side, which cannot be right. Practise this on a few examples until it feels automatic, because it is the step where most marks are lost on this topic.

A quick size check protects you. The hypotenuse is the longest side of the triangle. If you are finding it and your answer is smaller than another side, you multiplied when you should have divided. Use brackets on the calculator for the denominator, for example 0.9 ÷ sin 15, or press the fraction button and fill in the numerator and denominator separately.

Method: 1. Write the ratio with the unknown letter in it, for example sin 15^° = 0.9/h. 2. Swap the unknown and the ratio: h = frac0.9sin 15^°. 3. Check the calculator is in degree mode, then type it as a single division. 4. Compare with the other sides: the hypotenuse must be the longest.

Two-step problems and unrounded values

Some questions need two uses of trigonometry, or trigonometry followed by an area. In an isosceles triangle you split the shape into two right-angled triangles, find a height and half the base, then find the area. In a quadrilateral you may use one triangle to get a side that a second triangle needs.

Keep the unrounded value on the calculator, using the answer memory or storing it, so that the second step uses full accuracy. Rounding too early can change the last digit of the final answer. Round only at the end, to the number of decimal places or significant figures that the question asks for, and quote the unit.

Method: 1. Split the shape into right-angled triangles and mark the common side. 2. Find the shared side with the first trigonometry step and keep the full value. 3. Use that full value in the second step. 4. Round only the final answer and write the unit.

Bearings, 3D and harder set-ups (Higher)

On a Higher paper trigonometry shows up with bearings and with 3D shapes. A bearing is measured clockwise from north, so a ship that sails 18 km on a bearing of 050^° travels 18cos 50^° km north and 18sin 50^° km east. Draw a north line and mark the angle before you decide which side is opposite.

In a cuboid, the angle between a line and the base needs a right-angled triangle that includes the base diagonal. Find the diagonal with Pythagoras first, then use tangent with the vertical edge. In "elevation from two points" questions, write the height as h and make two equations with the same h.

Method: 1. Draw the situation, adding a north line or a base diagonal if needed. 2. Pick the right-angled triangle that contains the angle you want. 3. Find any missing side with Pythagoras or an earlier trigonometry step. 4. Use the inverse function for an angle, or rearrange for a side, and round at the end.