Trigonometric Ratios and Identities — SPI Formulas and Traps

Trigonometric Ratios of an Acute Angle

The six ratios of a right triangle

The six trigonometric ratios

\sin A = \frac{\text{opp}}{\text{hyp}},\ \cos A = \frac{\text{adj}}{\text{hyp}},\ \tan A = \frac{\text{opp}}{\text{adj}},\ \operatorname{cosec} A = \frac{\text{hyp}}{\text{opp}},\ \sec A = \frac{\text{hyp}}{\text{adj}},\ \cot A = \frac{\text{adj}}{\text{opp}}

oppthe side opposite the angle
adjthe side next to the angle (not the hypotenuse)
hypthe hypotenuse, across from the right angle

Finding every ratio from one given ratio

Reciprocal and quotient relations

\sin\theta \operatorname{cosec}\theta = 1,\quad \cos\theta \sec\theta = 1,\quad \tan\theta \cot\theta = 1,\quad \tan\theta = \frac{\sin\theta}{\cos\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta}

Dividing by cos to turn a fraction into tan

A sin and cos fraction in terms of tan

\frac{a\sin\theta + b\cos\theta}{c\sin\theta + d\cos\theta} = \frac{a\tan\theta + b}{c\tan\theta + d}

Common traps

Opposite and adjacent change with the angle

In a right triangle ABC with the right angle at B, side BC is opposite A but adjacent to C. Name the sides again every time the question moves to the other acute angle.

Reading sin A as sin times A

\sin A

is one number, the value of a ratio for the angle A. You cannot cancel the A in

\frac{\sin A}{A}

or split

\sin(A + B)

into

\sin A + \sin B

.

Pairing sec with sin

The names cross over: sec is the reciprocal of cos, and cosec is the reciprocal of sin. If

\cos\theta = \frac{3}{5}

, then

\sec\theta = \frac{5}{3}

.

Forgetting the square root

With hypotenuse 13 and one side 5, the third side is

\sqrt{169 - 25} = 12

, not 144. Take the root before you write the ratio.

Dividing only the top

Divide the numerator and the denominator by

\cos\theta

. Dividing only one of them changes the value of the fraction.

A plain number breaks the trick

In

\dfrac{\sin\theta + 1}{\cos\theta}

the 1 has no sin or cos, so dividing by

\cos\theta

leaves

\frac{1}{\cos\theta}

behind. Then you need the actual values of

\sin\theta

and

\cos\theta

from the sides.

Values at Standard Angles

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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

The table of standard values

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.

Common traps

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

is

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

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

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}

or

\frac{\theta}{3}

, finish the last division. Writing

\theta

itself is the most common slip on these questions.

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.

Trigonometric Identities

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sin²θ + cos²θ = 1 and its rearrangements

The first identity

\sin^2\theta + \cos^2\theta = 1, \qquad 1 - \sin^2\theta = \cos^2\theta, \qquad 1 - \cos^2\theta = \sin^2\theta

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

The second and third identities

1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \operatorname{cosec}^2\theta

Simplifying a square root of a ratio

Roots that clear for an acute angle A

\sqrt{\dfrac{1 + \cos A}{1 - \cos A}} = \operatorname{cosec} A + \cot A \qquad \sqrt{\dfrac{1 - \cos A}{1 + \cos A}} = \operatorname{cosec} A - \cot A \qquad \sqrt{1 - \sin^2 A} = \cos A \qquad \sqrt{\sec^2 A - 1} = \tan A

Using an identity as a second equation

The factored identities

(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1, \qquad (\operatorname{cosec}\theta + \cot\theta)(\operatorname{cosec}\theta - \cot\theta) = 1

Common traps

The identity needs the same angle

\sin^2 30^\circ + \cos^2 60^\circ = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

, not 1. The two angles must be equal for the sum to be 1.

The leftover goes with the bigger coefficient

5\sin^2\theta + 2\cos^2\theta

is

2 + 3\sin^2\theta

, not

2 + 3\cos^2\theta

. Take out the smaller coefficient and the remainder stays on the ratio that had more.

Getting the sign the wrong way round

\sec^2\theta - \tan^2\theta = 1

, so

\tan^2\theta - \sec^2\theta = -1

. Check which term comes first before you write 1.

Pairing sec with cot

The pairs are sec with tan and cosec with cot.

\sec^2\theta - \cot^2\theta

is not 1 for a general angle.

Taking the root of each term separately

\sqrt{1 - \sin^2 A}

is

\cos A

, not

1 - \sin A

. A root never splits across a plus or minus sign.

Plus and minus look alike in the options

The answer and a wrong option often differ only in one sign, such as

\operatorname{cosec} A + \cot A

and

\operatorname{cosec} A - \cot A

. Put in

A = 60^\circ

to tell them apart.

Trying to find the angle first

When

\sec\theta + \tan\theta = 3

, the angle is not a standard one. Use the reciprocal bracket and add; never hunt for

\theta

itself.

Adding when you need tan

Adding the two brackets cancels tan and gives

2\sec\theta

. Subtracting cancels sec and gives

2\tan\theta

. Choose the one that keeps the ratio you are asked for.

Ratios of Complementary Angles

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Switching a ratio with 90° minus the angle

Complementary angle rules

\sin(90^\circ - \theta) = \cos\theta,\quad \tan(90^\circ - \theta) = \cot\theta,\quad \sec(90^\circ - \theta) = \operatorname{cosec}\theta

Pairing terms of a long sum or product

Number of terms in a list of angles

\text{terms} = \frac{\text{last} - \text{first}}{\text{step}} + 1

Common traps

Keeping the same ratio

\sin(90^\circ - A)

is

\cos A

, not

\sin A

. The ratio always changes to its partner when the angle becomes

90^\circ

minus it.

Switching when the angles do not add to 90°

\sin 40^\circ

equals

\cos 50^\circ

, not

\cos 40^\circ

. Check that the two angles add to

90^\circ

before you swap the ratio.

Forgetting the unpaired terms

In a sum of

\sin^2

running up to

90^\circ

, the

45^\circ

term pairs with itself and the

90^\circ

term has no partner. Add

\frac{1}{2}

and 1 for them after counting the pairs.

Miscounting the terms

From

10^\circ

to

80^\circ

in steps of

10^\circ

there are 8 terms, not 7. Use last minus first, divided by the step, plus 1.

Trigonometric Ratios and Identities formulas

Trigonometric Ratios of an Acute Angle

The six ratios of a right triangle

Finding every ratio from one given ratio

Dividing by cos to turn a fraction into tan

Common traps

Values at Standard Angles

Reading the angle back from a value

Sides of a right triangle from one side and an angle

The table of standard values

Common traps

Trigonometric Identities

sin²θ + cos²θ = 1 and its rearrangements

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

Simplifying a square root of a ratio

Using an identity as a second equation

Common traps

Ratios of Complementary Angles

Switching a ratio with 90° minus the angle

Pairing terms of a long sum or product

Common traps

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