Triangle Trigonometry and Bearings

Two triangles with the same angles have sides in the same proportion, whatever their size. So once an angle is fixed, the ratio of any two sides is fixed too, and that ratio can be looked up instead of measured. Sine, cosine and tangent are the names of three such ratios.

What do sine, cosine and tangent actually mean?

In a right triangle, name the sides from the angle you are working with. The hypotenuse faces the right angle. The opposite side faces your angle, and the adjacent side lies beside it.

Sine is opposite ÷ hypotenuse, cosine is adjacent ÷ hypotenuse, and tangent is opposite ÷ adjacent. The memory aid SOHCAHTOA lists them in that order. See Sine, cosine and tangent.

The two acute angles of a right triangle add to 90°. The side opposite one of them is adjacent to the other, so $\sin \theta =\cos \left(90^{\circ }-\theta \right)$sin θ = cos(90° − θ). See Sine and cosine of complementary angles.

How do you find a missing side or angle?

1. Mark the hypotenuse. It always faces the right angle.
2. Mark opposite and adjacent from the angle in the question.
3. Write the ratio that links the side you know to the side you want, then solve.

If the hypotenuse is 10 and the angle is 30°, sine links them to the opposite side. See Solving right triangles.

When two sides are known and the angle is not, use the inverse function. ${\sin }^{-1}x$sin⁻¹ x is not $\frac{1}{\sin }x$1/sin x; it is the angle whose sine is x. See Finding an angle from a ratio.

An angle of elevation is measured up from the horizontal, and an angle of depression down from it. Because the two horizontals are parallel, the two angles are equal. See Elevation and depression.

Which values should you know without a calculator?

The values at 30°, 45° and 60°, because they come from two triangles you can draw. Cut a square of side 1 corner to corner for the 45° triangle.

Cut an equilateral triangle of side 2 in half for the 30° and 60° triangle.

Keep these in surd form: $\frac{\sqrt{3}}{2}$√3/2 is exact, and a rounded decimal is not. See Exact sine, cosine and tangent at 30, 45 and 60°.

What happens past 90 degrees?

A right triangle cannot hold an obtuse angle, so past 90° the ratios are defined on a circle of radius 1 instead. Put a point on the circle at angle $\theta$θ from the positive x-axis: its height is $\sin \theta$sin θ and its distance across is $\cos \theta$cos θ.

Turn past 90° and the point sits at the same height as the point for $180^{\circ }-\theta$180° − θ, but on the other side of the center. An obtuse angle and its supplement have the same sine and opposite cosines. See Sine and cosine of angles past 90 degrees.

What do the trigonometric graphs look like?

Plot the height of the point against the angle turned through. The graph of sine is a wave from 1 down to −1 that repeats every 360°. Cosine is the same wave slid 90° along, starting from 1. See The graphs of sine and cosine.

Tangent is $\sin \theta /\cos \theta$sin θ / cos θ. At 90° and 270°, $\cos \theta =0$cos θ = 0, so the graph has a vertical asymptote. Tangent repeats every 180° and takes every value. See The graph of tangent.

In y = a sin(b(x − c)) + d, the amplitude a is the distance from the principal axis to the crest, the period is 360° ÷ b, the phase shift c slides the wave sideways, and d is the height of the principal axis. See Amplitude and the principal axis and The period and phase shift of a wave.

How do you solve a trigonometric equation?

Find one solution with the inverse function, then use the symmetry of the graph for the rest. The sine curve is symmetric about 90°, so 180° − x is a solution too; cosine pairs x with 360° − x instead. See Solving simple trigonometric equations.

$2{\sin }^{2}x-\sin x-1=0$2 sin²x − sin x − 1 = 0 is a quadratic in sin x. Write s = sin x, factor, and solve sin x = each root separately, rejecting any root outside −1 to 1. See Quadratic equations in sine, cosine or tangent.

If the equation is in 2x, widen the interval before solving. For $0^{\circ }\le x\le 360^{\circ }$0° ≤ x ≤ 360°, 2x runs from 0° to 720°, so collect every solution for 2x in that range and then halve each one. See Trigonometric equations with a multiple angle.

The depth of water in a harbor rises and falls on a fixed cycle, so it fits the same equation. See Modeling tides and daylight with a sine function.

What do you do when the triangle has no right angle?

Use the sine rule or the cosine rule. In any triangle, $\frac{a}{\sin }A=\frac{b}{\sin }B=\frac{c}{\sin }C$a/sin A = b/sin B = c/sin C: each side divided by the sine of the angle facing it gives the same value. Use the sine rule when you know one side, the angle facing it, and one more side or angle. See The sine rule.

The cosine rule, $a^2=b^2+c^2-2$a² = b² + c² − 2bc cos A, starts the problems the sine rule cannot: two sides with the angle between them, or all three sides. At A = 90°, cos A = 0, the correction term vanishes, and Pythagoras is left. See The cosine rule.

With three sides known, rearrange for the cosine: $\cos A=\left(b^2+c^2-a^2\right)/2$cos A = (b² + c² − a²) / 2bc. See Finding angles with the cosine rule and Finding angles with the sine rule.

The area of any triangle is ½ab sin C, because the height onto a is b sin C. See Area with sine.

Why does the sine rule sometimes give two triangles?

Because an angle and its supplement have the same sine, and the calculator gives only the acute one.

Sometimes both triangles exist. Take A = 20° and $\sin B=\frac{1}{2}$sin B = 1/2. With B = 30° the third angle is 130°; with B = 150° it is 10°. Both fit. The check is whether the obtuse candidate leaves the third angle above zero: with A = 75° it does not, so only B = 30° fits. See The ambiguous case of the sine rule.

The cosine rule never has this problem. Cosine is negative for an obtuse angle and positive for an acute one, so the sign of the answer says which kind it is.

How do bearings work?

A bearing is an angle measured clockwise from north, written with three figures. North is 000°, east 090°, south 180° and west 270°, and 47° is written 047°. See Three-figure bearings.

The eight-point compass names north, east, south, west and the four directions between them. Northeast is 045° and southwest is 225°. See The eight-point compass.

A back bearing is the direction of the return journey. The north lines at both ends are parallel, so add 180° when the outward bearing is under 180° and subtract 180° when it is over. See Back bearings.

A navigation problem is usually a sine-rule or cosine-rule triangle. Convert the bearings to angles inside the triangle first. See Bearings with the sine and cosine rules.

How does trigonometry work in three dimensions?

Find the flat right triangle inside the solid and work in that. In a box, one round of Pythagoras gives the floor diagonal and a second round gives the space diagonal.

The angle between a line and a plane is measured to the projection of the line on the plane, so it lives in that same triangle. See Trigonometry in three dimensions, or solve the same problems with vectors.

The mistakes worth naming

• Labeling opposite and adjacent before choosing the angle. Only the hypotenuse is fixed.
• Reading ${\sin }^{-1}x$sin⁻¹ x as $\frac{1}{\sin }x$1/sin x. It is the angle whose sine is x.
• Giving one solution when the interval holds several. Use the symmetry of the graph to find the rest.
• Using the sine rule for an angle that may be obtuse. Check the supplement, or use the cosine rule.
• Halving a multiple-angle interval too early. Widen the interval, solve, then halve.

Learn this properly in the app

Every lesson linked above is a screen in Math Challenge with a diagram and practice questions. The right-angle groundwork is in lines, angles and Pythagoras, the same problems in space are in vectors, and identities and radian measure are in radians and trigonometric identities.