SPI Mathematics · Trigonometric Ratios and Identities

Values at Standard Angles

The ratios of 0°, 30°, 45°, 60° and 90° are known exact values, so an expression in them is plain arithmetic and an equation like sin 2θ = 1/2 gives the angle at once.

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Why this matters

Thirteen questions in the bank, the largest page in the chapter, and twelve are EASY. Six are arithmetic with these values, five give a value and ask for the angle, and two find the sides of a right triangle or a rectangle. All of them need the table on this page by heart.

Concept 1 of 3: The table of standard values

Learn the sine row as

\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}

for

0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ

. Cosine is the same list backwards. Tangent is sine divided by cosine, and the last three ratios are reciprocals.

Definition

$\sin 0^\circ = 0$ and $\cos 0^\circ = 1$ ; $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$ .
At $90^\circ$ : $\operatorname{cosec} 90^\circ = 1$ and $\cot 90^\circ = 0$ , while $\tan 90^\circ$ and $\sec 90^\circ$ are not defined.
$\tan 45^\circ = 1$ , and $\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}$ .
Not defined: $\tan 90^\circ$ , $\sec 90^\circ$ , $\cot 0^\circ$ and $\operatorname{cosec} 0^\circ$ , because each divides by zero.
$\sin^2 30^\circ$ means $(\sin 30^\circ)^2 = \frac{1}{4}$ .

Ratio	0°	30°	45°	60°
sin	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$
cos	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$
tan	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$
cosec	not defined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$
sec	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$
cot	not defined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$ At $90^\circ$ , $\tan 90^\circ$ and $\sec 90^\circ$ are not defined. They are not 0 and not 1.

At 90°: sin 90° = 1, cos 90° = 0, cosec 90° = 1, cot 90° = 0, and tan 90° and sec 90° are not defined. Sine rises from 0 to 1, cosine falls from 1 to 0, and tangent is their quotient.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Trigonometric Ratios and Identities · Values at Standard Angles

The value of

\dfrac{\sin 30 - 2\tan 45 + \cos 60}{\sin 45 \cos 45 + 2\sin 30 \cos 60} =

Doubling the angle does not double the value

\sin 60^\circ = \frac{\sqrt{3}}{2}

, but

2\sin 30^\circ = 1

. The ratio of twice an angle is not twice the ratio.

Squaring the angle instead of the value

\cos^2 60^\circ

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

. The square belongs to the value of the ratio, never to the angle.

Drill 5 more on the table of standard values

Concept 2 of 3: Reading the angle back from a value

The table works both ways. If

\sin(\text{something}) = \frac{\sqrt{3}}{2}

, that something is

60^\circ

. Find the whole angle inside the bracket first, then solve for

\theta

, then answer exactly what the question asks.

Definition

Match the value to the table: $\cos x = \frac{1}{2}$ gives $x = 60^\circ$ , $\tan x = 1$ gives $x = 45^\circ$ .
If the bracket holds $2\theta$ or $3\theta$ , solve for that first, then divide.
The angles here are taken from $0^\circ$ to $90^\circ$ , so each value gives one angle: $\tan\theta = 0$ means $\theta = 0^\circ$ .
Once the angle is known, any other ratio of it comes from the table.

Values that give the angle at once (0° to 90°)

\sin x = \tfrac{1}{2} \Rightarrow x = 30^\circ \qquad \cos x = \tfrac{1}{2} \Rightarrow x = 60^\circ \qquad \tan x = 1 \Rightarrow x = 45^\circ \qquad \tan x = \sqrt{3} \Rightarrow x = 60^\circ \qquad \tan x = \tfrac{1}{\sqrt{3}} \Rightarrow x = 30^\circ

Worked example

2\cos 2\theta = 1

, find

\theta

and then

\sin\theta

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Trigonometric Ratios and Identities · Values at Standard Angles

\sin 4\theta = \frac{\sqrt{3}}{2}

, then

\frac{\theta}{3}

equals:

Stopping at the angle in the bracket

From

\tan 2\theta = \sqrt{3}

you get

2\theta = 60^\circ

, so

\theta = 30^\circ

. The bracket's angle,

60^\circ

, is usually printed as a wrong option.

Answering the angle when a part of it is asked

If the question wants

\frac{\theta}{2}

\frac{\theta}{3}

, finish the last division. Writing

\theta

itself is the most common slip on these questions.

Drill 4 more on reading the angle back from a value

Concept 3 of 3: Sides of a right triangle from one side and an angle

If you know the hypotenuse and an angle, each leg is the hypotenuse times a ratio: the side opposite the angle uses sin, the side next to it uses cos. If you know a leg instead, tan links the two legs.

Definition

Opposite side $=$ hypotenuse $\times \sin\theta$ .
Adjacent side $=$ hypotenuse $\times \cos\theta$ .
Opposite side $=$ adjacent side $\times \tan\theta$ .
A right triangle with a $45^\circ$ angle has two equal legs. A rectangle's diagonal splits it into two right triangles, so its sides come the same way.

Legs from the hypotenuse

\text{opp} = \text{hyp} \times \sin\theta, \qquad \text{adj} = \text{hyp} \times \cos\theta, \qquad \text{opp} = \text{adj} \times \tan\theta

Worked example

A right triangle has a hypotenuse of 10 cm and one angle of

30^\circ

. Find its two legs and its area.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Trigonometric Ratios and Identities · Values at Standard Angles

In a right

\Delta XYZ

, XZ is the hypotenuse of length 12 cm and

\angle X = 45^\circ

. The area of the triangle is

Using sin for the side next to the angle

The side touching the angle (not the hypotenuse) is hypotenuse

\times \cos\theta

. Sin gives the side across from the angle. Swapping them swaps the two legs.

Taking the hypotenuse as a leg

The hypotenuse is always across from the right angle. In a right triangle PQR with hypotenuse PR, the right angle is at Q, and the area uses the two legs PQ and QR.

Drill 1 more on sides of a right triangle from one side and an angle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Reading the angle back from a value
Values that give the angle at once (0° to 90°)
$\sin x = \tfrac{1}{2} \Rightarrow x = 30^\circ \qquad \cos x = \tfrac{1}{2} \Rightarrow x = 60^\circ \qquad \tan x = 1 \Rightarrow x = 45^\circ \qquad \tan x = \sqrt{3} \Rightarrow x = 60^\circ \qquad \tan x = \tfrac{1}{\sqrt{3}} \Rightarrow x = 30^\circ$
Sides of a right triangle from one side and an angle
Legs from the hypotenuse
$\text{opp} = \text{hyp} \times \sin\theta, \qquad \text{adj} = \text{hyp} \times \cos\theta, \qquad \text{opp} = \text{adj} \times \tan\theta$

Reference tables (1)

The table of standard values6 rows

Ratio	0°	30°	45°	60°
sin	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$
cos	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$
tan	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$
cosec	not defined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$
sec	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$
cot	not defined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$ At $90^\circ$ , $\tan 90^\circ$ and $\sec 90^\circ$ are not defined. They are not 0 and not 1.

At 90°: sin 90° = 1, cos 90° = 0, cosec 90° = 1, cot 90° = 0, and tan 90° and sec 90° are not defined. Sine rises from 0 to 1, cosine falls from 1 to 0, and tangent is their quotient.

Watch out for (6)

Doubling the angle does not double the value
→ The table of standard values
Squaring the angle instead of the value
→ The table of standard values
Stopping at the angle in the bracket
→ Reading the angle back from a value
Answering the angle when a part of it is asked
→ Reading the angle back from a value
Using sin for the side next to the angle
→ Sides of a right triangle from one side and an angle
Taking the hypotenuse as a leg
→ Sides of a right triangle from one side and an angle

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