Trigonometry – sine, cosine and tangent

What are sin, cos and tan?

In a right-angled triangle with angle u:

sin(u) = opposite / hypotenuse

cos(u) = adjacent / hypotenuse

tan(u) = opposite / adjacent = sin(u) / cos(u)

Memory aid — SOH-CAH-TOA:

Sin = Opposite / Hypotenuse (opposite/hypotenuse)
Cos = Adjacent / Hypotenuse (adjacent/hypotenuse)
Tan = Opposite / Adjacent (opposite/adjacent)

Say it aloud a few times to make it easier to remember.

Remember: these definitions only apply in right-angled triangles for angles from 0° to 90°. For other angles (over 90°, negative, or in non-right-angled triangles), we need the unit circle, and the rules the sine rule and the cosine rule.

Important values

Degrees	cos	sin	tan
0°	1	0	0
30°	√3 / 2	1 / 2	√3 / 3
45°	√2 / 2	√2 / 2	1
60°	1 / 2	√3 / 2	√3
90°	0	1	undefined
120°	−1 / 2	√3 / 2	−√3
135°	−√2 / 2	√2 / 2	−1
150°	−√3 / 2	1 / 2	−√3 / 3
180°	−1	0	0

Why «undefined»? tan(90°) = sin(90°)/cos(90°) = 1/0, and we cannot divide by 0. The same applies to tan(270°). This gives vertical asymptotes in the tan graph.

Rationalise the denominator! Write tan(30°) = √3/3, not 1/√3. Both expressions have the same value; here we write the answer with a rational denominator.

Memory aid: The cos values go «down» (1 → 0), while the sin values go «up» (0 → 1) from 0° to 90°. They are mirror images of each other. From 90° to 180°, sin decreases again and cos becomes negative — hence the minus signs above 90°.

Range

Which values can sin, cos and tan take?

Function	Range	Period
sin x	−1 ≤ y ≤ 1	360°
cos x	−1 ≤ y ≤ 1	360°
tan x	all real numbers (but undefined at 90° + n·180°)	180°

Consequence: The equation sin(x) = 1.5 has no solution. Sin can never be greater than 1.

The unit circle

The unit circle is a circle with radius 1 and its centre at the origin.

For an angle u from the positive x-axis:

the x coordinate of the point on the circle = cos(u)
the y coordinate of the point on the circle = sin(u)

Therefore, we always have: cos²(u) + sin²(u) = 1

The unit circle lets us extend sin and cos to all angles (including angles above 90° and negative angles).

Where does sin² + cos² = 1 come from?

Directly from Pythagoras' theorem! The point (cos u, sin u) lies on the unit circle, a distance of 1 from the origin. The vertical side is sin u, the horizontal side is cos u, and the hypotenuse is 1.

Pythagoras: cos²(u) + sin²(u) = 1² = 1

Practical use: If you know sin(u) = 0.6, you can find cos(u) without calculating the angle: cos²(u) = 1 − 0.36 = 0.64 ⇒ cos(u) = ±0.8 (depending on the quadrant). More about Pythagoras.

Sign rule for the four quadrants

As the angle increases from 0° to 360°, the point moves through four quadrants. The signs of sin, cos and tan vary between quadrants:

Quadrant	Angle	sin	cos	tan
In	0°–90°	+	+	+
II	90°–180°	+	−	−
III	180°–270°	−	−	+
IV	270°–360°	−	+	−

Memory aid — «All Sin Tan Cos»:

All (I): all three are positive
Sin (II): only sin is positive
Tan (III): only tan is positive
Cos (IV): only cos is positive

Go anticlockwise around the circle and read «A-S-T-C».

The sine rule

In every triangle:

a / sin(A) = b / sin(B) = c / sin(C)

Each side divided by the sine of its opposite angle gives the same number.

Use it when you know one side and two angles (ASA), or two sides and an opposite angle.

Which form should you use?

Both forms below are mathematically equivalent — one is simply inverted. But one makes the work easier depending on what you are looking for:

Find a SIDE
a / sin(A) = b / sin(B)

Sides in the numerator of the fraction. You multiply by sin and isolate the side in one step.

Example: knowing A, B, b — find a.
a = b · sin(A) / sin(B)

Find an ANGLE
sin(A) / a = sin(B) / b

Angles in the numerator of the fraction. You isolate sin(B) directly and use inverse sine.

Example: knowing a, A, b — find B.
sin(B) = b · sin(A) / a, so B = sin⁻¹(…)

Memory aid: you want what you are looking for in the numerator of the fraction. This avoids having to cross-multiply first.

The cosine rule

A generalisation of Pythagoras' theorem for all triangles:

c² = a² + b² − 2ab · cos(C)

When C = 90° → cos(90°) = 0, and we get Pythagoras' theorem: c² = a² + b².

Use it when you know three sides (SSS) or two sides and the included angle (SAS).

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