Trigonometry basics

Sine, cosine and tangent of an acute angle, the values for special angles, the unit circle and the basic trigonometric identity.

In this lesson you will learn

Define sin, cos and tan in a right triangle
Know the values for 30°, 45° and 60°
Calculate an unknown side or height
See sine and cosine on the unit circle

How can you measure the height of the Maiden Tower in Baku without climbing it? It is enough to stand some distance away, measure the angle up to the top and use trigonometry. Trigonometry studies how the angles and sides of a triangle are related, and it is used every day in engineering, astronomy and navigation.

Sine, cosine and tangent in a right triangle

Take an acute angle α in a right triangle. The leg across from the angle is the opposite side, and the leg that forms the angle is the adjacent side. The ratios of the sides depend only on the angle, not on the size of the triangle, which is why they have special names.

sin α = a / c cos α = b / c tan α = a / bsin α = a / c cos α = b / c tan α = a / b

where:

aopposite side — the leg across from α
badjacent side — the leg that forms α
chypotenuse

Also, tan α = sin α / cos α. Because the hypotenuse is the longest side, for an acute angle 0 < sin α < 1 and 0 < cos α < 1.

The 3, 4, 5 triangle

A right triangle has legs 3 and 4 and hypotenuse 5. Find the sine, cosine and tangent of the angle α opposite the leg of length 3.

Show solution

Opposite side 3, adjacent side 4, hypotenuse 5.
sin α = 3/5 = 0.6
cos α = 4/5 = 0.8
tan α = 3/4 = 0.75

Special angles

α	30°	45°	60°
sin α	1/2	√2/2	√3/2
cos α	√3/2	√2/2	1/2
tan α	√3/3	1	√3

These values come from half of an equilateral triangle (30°, 60°) and the diagonal of a square (45°). As sin α grows, cos α shrinks.

The height of the Maiden Tower

Elvin stands 40 m from the tower and sees its top at an angle of 35°. His eyes are 1.5 m above the ground. How tall is the tower? (tan 35° ≈ 0.70)

Show solution

The height from eye level to the top is the opposite side, and 40 m is the adjacent side.
tan 35° = x / 40 ⇒ x = 40 · tan 35° ≈ 40 · 0.70 = 28 m.
Add Elvin's eye height: 28 + 1.5 = 29.5 m, which matches the tower's known height.

A ladder

A 6 m ladder makes an angle of 60° with the ground. How high is its top?

Show solution

The ladder is the hypotenuse, and the height is the side opposite the 60° angle.
h = 6 · sin 60° = 6 · √3/2 = 3√3 ≈ 5.2 m.

The basic identity and the unit circle

sin²α + cos²α = 1

This identity follows from the Pythagorean theorem. For example, if sin α = 0.6 for an acute angle, then cos α = √(1 − 0.36) = √0.64 = 0.8.

A circle with its centre at the origin and radius 1 is called the unit circle. A point on it, turned by an angle α from the x-axis, has coordinates (cos α, sin α). This definition gives sine and cosine for any angle, even angles larger than 90°: for example, cos 120° = −1/2.

Interactive

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Drag the point around the circle: its x-coordinate is cos α and its y-coordinate is sin α. Check the table values for 30°, 45° and 60°.

Key points

Sine and cosine are a leg divided by the hypotenuse, tangent is one leg divided by the other: sin α = a/c, cos α = b/c, tan α = a/b.
sin 30° = cos 60° = 1/2, sin 45° = cos 45° = √2/2, tan 45° = 1.
Basic identity: sin²α + cos²α = 1.
On the unit circle, a point has coordinates (cos α, sin α).

Check yourself

10 questions. Every correct answer earns XP.

1 / 10

In a right triangle, what is the sine of an acute angle α?