Trigonometric ratios

Q: How do I remember sin, cos and tan?

SOH-CAH-TOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

Q: What are the trigonometric values of standard angles?

See the table above for 0°, 30°, 45°, 60° and 90°. Cos is sin read backwards, and \tan\theta = \sin\theta \div \cos\theta.

Q: What are angle of elevation and angle of depression?

Elevation is the angle above the horizontal when you look up at an object. Depression is the angle below the horizontal when you look down at it.

Q: Is tan 90° defined?

No. At 90° the adjacent side is 0, and dividing by 0 is not allowed.

Trigonometric ratios compare the sides of a right-angled triangle. For an acute angle $θ$ , the side opposite it, the side next to it (adjacent) and the hypotenuse give the three ratios sine, cosine and tangent.

Formulas

The three ratios

sin θ = \frac{opposite}{hypotenuse}, cos θ = \frac{adjacent}{hypotenuse}, tan θ = \frac{opposite}{adjacent}

Basic identity

sin^{2} θ + cos^{2} θ = 1 tan θ = \frac{sin θ}{cos θ}

Values of sin at the standard angles

sin 0^{\circ} = 0, sin 3 0^{\circ} = \frac{1}{2}, sin 4 5^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = \frac{3}{2}, sin 9 0^{\circ} = 1

Cos takes the same values in reverse order

cos 0^{\circ} = 1, cos 3 0^{\circ} = \frac{3}{2}, cos 4 5^{\circ} = \frac{1}{2}, cos 6 0^{\circ} = \frac{1}{2}, cos 9 0^{\circ} = 0

Worked examples

Example 1. In a right-angled triangle, the side opposite angle

θ

is 3, the adjacent side is 4 and the hypotenuse is 5. Find the three ratios.

$sin θ = \frac{3}{5}$ .
$cos θ = \frac{4}{5}$ .
$tan θ = \frac{3}{4}$ .

Answer:

\frac{3}{5}, \frac{4}{5}, \frac{3}{4}

Example 2. Find the value of

sin 3 0^{\circ} + cos 6 0^{\circ}

$sin 3 0^{\circ} = \frac{1}{2}$ and $cos 6 0^{\circ} = \frac{1}{2}$ .
Sum = $\frac{1}{2} + \frac{1}{2} = 1$ .

Answer: 1

Example 3. The angle of elevation of the top of a tower is

4 5^{\circ}

from a point 30 m from its foot. Find the height of the tower.

$tan 4 5^{\circ} = \frac{height}{30}$ and $tan 4 5^{\circ} = 1$ .
Height = $30 \times 1 = 30$ .

Answer: 30 m

Example 4. A 10 m ladder makes an angle of

6 0^{\circ}

with the ground. How high up the wall does it reach?

$sin 6 0^{\circ} = \frac{height}{10}$ .
Height = $10 \times \frac{3}{2} = 53 \approx 8.66$ .

Answer:

53 \approx 8.66 m

Common mistakes

Mixing up opposite and adjacent. They are named relative to the angle you are working with.
Using SOH-CAH-TOA on a triangle with no right angle.
Having the calculator in radians instead of degrees.

Practice questions with answers

1. What is sin 30°?

A $\frac{1}{2}$
B $\frac{3}{2}$
C1
D $\frac{1}{2}$

Show answer

Answer: A)

\frac{1}{2}

Standard value: sin 30° =

\frac{1}{2}

2. What is the sine of 45°?

A $\frac{1}{2}$
B $\frac{1}{2}$
C $\frac{3}{2}$
D1

Show answer

Answer: B)

\frac{1}{2}

The sine of 45° is

\frac{1}{2}

, a standard angle value.

3. tan 45° is equal to:

Show answer

Answer: A) 1

tan 45° = sin 45 / cos

45 = \frac{1}{2} / \frac{1}{2} = 1

4. If the angle of elevation of the Sun increases, the length of the shadow of a vertical pole:

Adecreases
Bincreases
Cstays the same
Ddoubles

Show answer

Answer: A) decreases

Shadow =

\frac{h e i g h t}{t an ( t h e t a )}

. As θ increases, tan(θ) increases, so the shadow length decreases.

5. As θ increases from 0° towards 90°, the value of tan θ:

Adecreases steadily from 1 to 0
Bincreases from 0 up to 1 and stops there
Cstays constant at 1 throughout
Dincreases from 0 and grows without any upper limit, and is undefined at 90°

Show answer

Answer: D) increases from 0 and grows without any upper limit, and is undefined at 90°

tan θ =

\frac{s i n θ}{c o s θ}

. As θ grows, sin θ rises towards 1 while cos θ falls towards 0, so the ratio keeps getting larger (tan 45° = 1, tan 60° =

3

≈ 1.73, tan 89° ≈ 57). At 90° the denominator is 0, so tan 90° is undefined.

6. Assertion (A): For 0° < θ < 90°, the value of sec θ is always greater than or equal to 1. Reason (R): sec θ =

\frac{1}{c o s θ}

and cos θ lies between 0 and 1 for these angles. Which option is correct?

ABoth A and R are true and R is the correct explanation of A.
BBoth A and R are true but R is not the correct explanation of A.
CA is true but R is false.
DA is false but R is true.

Show answer

Answer: A) Both A and R are true and R is the correct explanation of A.

For 0° ≤ θ < 90°, cos θ is between 0 and 1, so its reciprocal sec θ is at least 1. R correctly explains A.

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Frequently asked questions

How do I remember sin, cos and tan?

SOH-CAH-TOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

What are the trigonometric values of standard angles?

See the table above for 0°, 30°, 45°, 60° and 90°. Cos is sin read backwards, and $tan θ = sin θ \div cos θ$ .

What are angle of elevation and angle of depression?

Elevation is the angle above the horizontal when you look up at an object. Depression is the angle below the horizontal when you look down at it.

Is tan 90° defined?

No. At 90° the adjacent side is 0, and dividing by 0 is not allowed.

Trigonometric ratios

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics