Draw the two triangles.
Every value on this page comes out of these.
Then it is only SOH-CAH-TOA on a triangle whose sides you already know.
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Exact Trig Values
Nine values for 30°, 45° and 60° — all of them read off two triangles.
The problem
Give the exact value of sin 60° — no decimals.
The 30-60-90 triangle has sides 1, √3 and 2. At 60° the opposite side is √3 and the hypotenuse is 2, so sin 60° = √32. D is sin 30° — the two trade places depending on which angle you stand at.
Read the values off.
Standing at the 30° corner: opposite 1, adjacent √3, hypotenuse 2.
sin 30 = opp/hyp = ½
1 over 2
cos 30 = adj/hyp = √32
√3 over 2
tan 30 = opp/adj = 1√3
rationalise it
= √33
the usual written form
Move to the 60° corner of the same triangle and opposite and adjacent swap over — which is why sin 60 and cos 30 come out identical.
The whole table.
| 30° | 45° | 60° | |
|---|---|---|---|
| sin | ½ | √22 | √32 |
| cos | √32 | √22 | ½ |
| tan | √33 | 1 | √3 |
Read the sine row left to right and it climbs; the cosine row is the same numbers backwards. They are mirror images because 30° and 60° are the two acute angles of one triangle.
The pattern that makes it memorable.
Write √0, √1, √2, √3, √4 over 2 and the whole sine row appears.
| Angle | Pattern | sin |
|---|---|---|
| 0° | √02 | 0 |
| 30° | √12 | ½ |
| 45° | √22 | √22 |
| 60° | √32 | √32 |
| 90° | √42 | 1 |
Cosine runs the same list in reverse, and tan is simply sin ÷ cos.
tan 60 = sin 60 ÷ cos 60
the quotient rule
= (√32) ÷ (12)
the halves cancel
= √3
≈ 1.73
Why exact beats decimal.
exact
√32
no rounding, ever
decimal
0.866…
errors build up
Answer choices for these questions are printed in surd form, so a decimal answer will not match anything on the page.
The half that catches people
sin 30 = ½, not √32. The smaller angle faces the shorter side, so it gets the smaller value.
Variations: 0° and 90°, squares inside an expression, a tan equation.
Evaluate sin 90° + cos 0° − tan 0°, and find tan 90°.
sin 90° = 1, cos 0° = 1, tan 0° = 0
ends of the √0 … √4 pattern
1 + 1 − 0 = 2
the expression
tan 90° = 1 ÷ 0
sin 90° ÷ cos 90°
undefined
no number at all
Different: at 0° and 90° the triangle flattens, so read the values from the pattern. tan 90° is undefined, not 0 and not 1.
Evaluate 4 sin² 60° − 2 cos² 45°.
sin² 60° = (√32)² = 34
square the value, not the angle
cos² 45° = (√22)² = 24 = 12
the root disappears
4(34) − 2(12) = 3 − 1 = 2
multiply after squaring
sin² 60° is (sin 60°)²
It is not sin(60²) and not 2 sin 60°. Squaring √32 gives 34; doubling it gives √3, the classic wrong line that turns this answer into 4√3 − 2√2 instead of 2.
θ is acute and 3 tan θ = √3. Find cos θ.
tan θ = √33
÷ 3
θ = 30°
the tan row: √33, 1, √3
cos θ = √32
the ratio actually asked
Different: the given value is a tan, so look along the tan row — √33 is the same number as 1√3.
Using the table both ways.
Backwards: θ is acute and √2 cos θ − 1 = 0. Find tan θ. Solve for the ratio, find it in the table, read off the angle.
cos θ = 1√2 = √22
+ 1, ÷ √2, rationalise
θ = 45°
the cos row of the table
tan θ = 1
the question’s real ask
Inside a figure: a rectangle’s diagonal is 10 and makes a 60° angle with the shorter side. Find the area. The diagonal is the hypotenuse of a right triangle whose legs are the two sides.
shorter = 10 cos 60° = 10 × ½ = 5
adjacent to the 60°
longer = 10 sin 60° = 10 × √32 = 5√3
opposite the 60°
area = 5 × 5√3 = 25√3
length × width
Which side touches the angle?
The side the angle sits on is adjacent (use cos); the side across from it is opposite (use sin). Swapping them does not change this area, but it does change any question that asks for one side.
An angle of depression. From the top of a 30 m tower the angle of depression to a point on the ground is 60°. The horizontal at the top is parallel to the ground, so the angle of elevation at the point is also 60°. The tower is opposite it and the distance d adjacent: tan 60° = 30d, so d = 30 ÷ √3 = 30√33 = 10√3 m. The unknown is underneath, so we divide by the tan value.
On the exam
Expect three shapes of question: evaluate an expression of exact values (square carefully — tan² 60° is 3), solve a simple equation and name the angle, or find sides of a figure with a 30°, 45° or 60° angle. Write each value from the table before you combine anything. Ten-second check: sin² + cos² = 1 for any angle you used.
Check: the Pythagorean identity.
sin²θ + cos²θ must equal 1 for every angle.
at 30°
¼ + ¾ = 1
✓
at 45°
½ + ½ = 1
(√22)² = ½ ✓
at 60°
¾ + ¼ = 1
✓
Now you try
Practice problem
Evaluate exactly
sin 30° + cos 60°
sin 30° = ½ and cos 60° = ½, so the total is 1. They match because 30° and 60° are complementary — the sine of an angle always equals the cosine of what is left over from 90°.