Mathematics · Foundational · 35 min · Chapter 4 of 28

Trigonometry

Sine and cosine as coordinates on the unit circle, the identities that follow, and the laws that solve any triangle.

The idea

In a right triangle, sin θ is opposite over hypotenuse, cos θ is adjacent over hypotenuse, and tan θ is opposite over adjacent. Those ratios depend only on the angle, not on the size of the triangle, which is why they are functions of θ alone. The unit-circle definition keeps the same values and extends them to every angle, including those that do not fit inside a right triangle: negative angles, reflex angles, and angles beyond a full turn.

Once sine and cosine are coordinates, identities are geometry. The point (cos θ, sin θ) lies on x² + y² = 1, so sin²θ + cos²θ = 1. Dividing by cos²θ produces the companion 1 + tan²θ = sec²θ. Adding angles corresponds to multiplying complex numbers on the unit circle, or to a rotation matrix; either route yields sin(α + β) = sin α cos β + cos α sin β and cos(α + β) = cos α cos β − sin α sin β.

For a general triangle, label sides a, b, c opposite angles A, B, C. The law of sines says the sides are proportional to the sines of the opposite angles, with common ratio equal to the diameter of the circumscribed circle. The law of cosines is the Pythagorean theorem with a correction −2bc cos A. Use sines when you know an angle and its opposite side; use cosines when you know three sides, or two sides and the included angle.

Hold onto this

On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate of the point at angle θ.
The identity sin²θ + cos²θ = 1 is Pythagoras in that circle.
Right triangles give the first definitions; the laws of sines and cosines extend them to every triangle.
Angle sum formulas are how you turn a shift of angle into algebra.

Formulas

Right-triangle ratios

sin θ = \frac{opp}{hyp}, cos θ = \frac{adj}{hyp}, tan θ = \frac{opp}{adj}

Definitions in a right triangle. The angle θ is one of the acute angles, not the right angle.

opp, adj: Sides relative to the chosen acute angle
hyp: Side opposite the right angle

Pythagorean identity

sin^{2} θ + cos^{2} θ = 1

True for every angle. It is the unit circle written in function notation.

θ: Any real angle

Law of sines

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Sides are proportional to the sines of the opposite angles.

a, b, c: Side lengths
A, B, C: Opposite angles

Law of cosines

c^{2} = a^{2} + b^{2} - 2 ab cos C

Pythagoras, corrected by the included angle. If C = 90°, cos C = 0 and the correction vanishes.

C: The angle between sides a and b
c: The side opposite C

Angle sum

sin (α + β) = sin α cos β + cos α sin β

Sine of a sum is not the sum of sines. The expansion mixes both functions.

α, β: Any two angles

Worked examples

1. A height from an angle

From 40 m away, the top of a mast makes a 28° angle with the horizontal. How tall is the mast, ignoring eye height?

Opposite side is the height h. Adjacent side is 40 m.
tan 28° = h / 40.
h = 40 tan 28°.
tan 28° ≈ 0.5317, so h ≈ 21.3 m.

About 21.3 meters.

2. A side from the law of cosines

A triangle has sides 7 and 9 with included angle 40°. Find the third side.

c² = 7² + 9² − 2·7·9·cos 40°.
c² = 49 + 81 − 126 cos 40°.
cos 40° ≈ 0.7660, so 126 × 0.7660 ≈ 96.5.
c² ≈ 130 − 96.5 = 33.5, so c ≈ 5.79.

The third side is about 5.79.

Try these

If sin θ = 3/5 and θ is acute, what is cos θ?

Why does the law of cosines become Pythagoras when the included angle is a right angle?

Common mistakes

Calculators in degree mode and radian mode answer different questions. Match the mode to the unit of the angle.
The law of sines can be ambiguous (SSA): a given sine corresponds to two supplementary angles. Check that the angles sum to 180°.
sin(α + β) is not sin α + sin β.