# Trigonometric Ratios Of Standard Angles

Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. The standard angles for these trigonometric ratios are 0°, 30°, 45°, 60° and 90°. These angles can also be represented in the form of radians such as 0, π/6, π/4, π/3, and π/2. These angles are most commonly and frequently used in trigonometry. Learning the values of these trigonometry angles is very necessary to solve various problems.

Trigonometric Ratios Formulas:

The six trigonometric ratios are basically expressed in terms of the right-angled triangle.

∆ABC is a right-angled triangle, right-angled at (shown in figure 1). The six trigonometric ratios for ∠C are defined as:

$$\begin{array}{l}sin~∠C\end{array}$$
=
$$\begin{array}{l}\frac{AB}{AC}\end{array}$$

$$\begin{array}{l}cosec~∠C\end{array}$$
=
$$\begin{array}{l}\frac{1}{sin~∠C}\end{array}$$

$$\begin{array}{l}cos~∠C\end{array}$$
=
$$\begin{array}{l}\frac{BC}{AC}\end{array}$$

$$\begin{array}{l}sec~∠C\end{array}$$
=
$$\begin{array}{l}\frac{1}{cos~∠C}\end{array}$$

$$\begin{array}{l}tan~∠C\end{array}$$
=
$$\begin{array}{l}\frac{sin~∠C}{cos~∠C}\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cot~∠C\end{array}$$
=
$$\begin{array}{l}\frac{1}{tan~∠C}\end{array}$$

The standard angles for which trigonometric ratios can be easily determined are

$$\begin{array}{l}0°,30°,45°,60°\end{array}$$
and
$$\begin{array}{l}90°\end{array}$$
. The values are determined using properties of triangles. The two acute angles of a right-angled triangle are complementary.

## Trigonometric Ratios Table (Standard Angles)

 Angle = ∠C 0° 30° 45° 60° 90° $$\begin{array}{l}sin~C\end{array}$$ 0 $$\begin{array}{l}\frac{1}{2}\end{array}$$ $$\begin{array}{l}\frac{1}{\sqrt{2}}\end{array}$$ $$\begin{array}{l}\frac{\sqrt{3}}{2}\end{array}$$ 1 $$\begin{array}{l}cos~C\end{array}$$ 1 $$\begin{array}{l}\frac{\sqrt{3}}{2}\end{array}$$ $$\begin{array}{l}\frac{1}{\sqrt{2}}\end{array}$$ $$\begin{array}{l}\frac{1}{2}\end{array}$$ 0 $$\begin{array}{l}tan~C\end{array}$$ 0 $$\begin{array}{l}\frac{1}{\sqrt{3}}\end{array}$$ 1 $$\begin{array}{l}\sqrt{3}\end{array}$$ Not Defined $$\begin{array}{l}cosec~C\end{array}$$ Not Defined 2 $$\begin{array}{l}\sqrt{2}\end{array}$$ $$\begin{array}{l}\frac{2}{\sqrt{3}}\end{array}$$ 1 $$\begin{array}{l}sec~C\end{array}$$ 1 $$\begin{array}{l}\frac{2}{\sqrt{3}}\end{array}$$ $$\begin{array}{l}\sqrt{2}\end{array}$$ 2 Not Defined $$\begin{array}{l}cot~C\end{array}$$ Not Defined $$\begin{array}{l}\sqrt{3}\end{array}$$ 1 $$\begin{array}{l}\frac{1}{\sqrt{3}}\end{array}$$ 0

The above table shows the important angles for all the six trigonometric ratios. Let us learn here how to derive these values.

## Derivation of Trigonometric Ratios for Standard Angles

Value of Trigonometric Ratios for Angle equal to 45 degrees

In

$$\begin{array}{l}⊿ABC\end{array}$$
, if
$$\begin{array}{l}∠C\end{array}$$
=
$$\begin{array}{l}45°\end{array}$$
, then
$$\begin{array}{l}∠A\end{array}$$
=
$$\begin{array}{l}45°\end{array}$$
. Since the angles are equal,
$$\begin{array}{l}⊿ABC\end{array}$$
becomes a right angled isosceles triangle. So,
$$\begin{array}{l}AB\end{array}$$
=
$$\begin{array}{l}BC\end{array}$$
. Assume
$$\begin{array}{l}AB\end{array}$$
=
$$\begin{array}{l}BC\end{array}$$
=
$$\begin{array}{l}a\end{array}$$
units.

Using Pythagoras theorem ,

$$\begin{array}{l}AC^2\end{array}$$
=
$$\begin{array}{l}AB^2~+~BC^2\end{array}$$

$$\begin{array}{l}AC^2\end{array}$$
=
$$\begin{array}{l}a^2~+~a^2\end{array}$$

$$\begin{array}{l}AC\end{array}$$
=
$$\begin{array}{l}a\sqrt{2} ~units\end{array}$$

$$\begin{array}{l}∠C\end{array}$$
=
$$\begin{array}{l}45°\end{array}$$

$$\begin{array}{l}∴~ sin~∠C\end{array}$$
=
$$\begin{array}{l}sin~45°\end{array}$$
=
$$\begin{array}{l}\frac{AB}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{a}{a\sqrt{2}}\end{array}$$
=
$$\begin{array}{l}\frac{1}{\sqrt{2}}\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cosec~45°\end{array}$$
=
$$\begin{array}{l}\frac{1}{sin~45°}\end{array}$$
=
$$\begin{array}{l}\sqrt{2}\end{array}$$

$$\begin{array}{l}cos~∠C\end{array}$$
=
$$\begin{array}{l}cos~45°\end{array}$$
=
$$\begin{array}{l}\frac{BC}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{a}{a\sqrt{2}}\end{array}$$
=
$$\begin{array}{l}\frac{1}{\sqrt{2}}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}sec~45°\end{array}$$
=
$$\begin{array}{l}\frac{1}{cos~45°}\end{array}$$
=
$$\begin{array}{l}\sqrt{2}\end{array}$$

$$\begin{array}{l}tan~45°\end{array}$$
=
$$\begin{array}{l}\frac{sin~45°}{cos~45°}\end{array}$$
=
$$\begin{array}{l}\frac{\frac{a}{\sqrt{2}}}{\frac{a}{\sqrt{2}}}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cot~45°\end{array}$$
=
$$\begin{array}{l}\frac{1}{tan~45°}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

Value of Trigonometric Ratios for Angle equal to 30 and 60 degrees

In figure 3,

$$\begin{array}{l}∆PQR\end{array}$$
is equilateral. The perpendicular from any vertex on the opposite side is coincident with the angle bisector of that particular vertex. Also, the perpendicular bisects the opposite side. If a perpendicular
$$\begin{array}{l}PS\end{array}$$
is dropped on
$$\begin{array}{l}QR\end{array}$$
, then
$$\begin{array}{l}∠QPS\end{array}$$
=
$$\begin{array}{l}∠SPR\end{array}$$
=
$$\begin{array}{l}30°\end{array}$$
and
$$\begin{array}{l}QS\end{array}$$
=
$$\begin{array}{l}SR\end{array}$$
. Assume
$$\begin{array}{l}PQ\end{array}$$
=
$$\begin{array}{l}QR\end{array}$$
=
$$\begin{array}{l}RP\end{array}$$
=
$$\begin{array}{l}2a\end{array}$$
units.

$$\begin{array}{l}⇒~QS\end{array}$$
=
$$\begin{array}{l}SR\end{array}$$
=
$$\begin{array}{l}a\end{array}$$
units

In

$$\begin{array}{l}∆PSQ\end{array}$$
, by Pythagoras theorem,

$$\begin{array}{l}PQ^2\end{array}$$
=
$$\begin{array}{l}QS^2~+~PS^2\end{array}$$

$$\begin{array}{l}PS^2\end{array}$$
=
$$\begin{array}{l}(2a)^2~-~a^2\end{array}$$

$$\begin{array}{l}PS\end{array}$$
=
$$\begin{array}{l}\sqrt{3a^2}\end{array}$$
=
$$\begin{array}{l}\sqrt{3} a\end{array}$$

$$\begin{array}{l}∠SPQ\end{array}$$
=
$$\begin{array}{l}30°\end{array}$$

$$\begin{array}{l}sin~∠SPQ\end{array}$$
=
$$\begin{array}{l}sin~30°\end{array}$$
=
$$\begin{array}{l}\frac{SQ}{PQ}\end{array}$$
=
$$\begin{array}{l}\frac{a}{2a}\end{array}$$
=
$$\begin{array}{l}\frac{1}{2}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cosec~30°\end{array}$$
=
$$\begin{array}{l}\frac{1}{sin~30°}\end{array}$$
=
$$\begin{array}{l}2\end{array}$$

$$\begin{array}{l}cos~∠SPQ\end{array}$$
=
$$\begin{array}{l}cos~30°\end{array}$$
=
$$\begin{array}{l}\frac{PS}{PQ}\end{array}$$
=
$$\begin{array}{l}\frac{\sqrt{3} a}{2a}\end{array}$$
=
$$\begin{array}{l}\frac{\sqrt{3}}{2}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}sec~30°\end{array}$$
=
$$\begin{array}{l}\frac{1}{cos~30°}\end{array}$$
=
$$\begin{array}{l}\frac{2}{\sqrt{3}}\end{array}$$

$$\begin{array}{l}tan~30°\end{array}$$
=
$$\begin{array}{l}\frac{sin~30°}{cos~30°}\end{array}$$
=
$$\begin{array}{l}\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\end{array}$$
=
$$\begin{array}{l}\frac{1}{\sqrt{3}}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cot~30°\end{array}$$
=
$$\begin{array}{l}\frac{BC}{AB}\end{array}$$
=
$$\begin{array}{l}\sqrt{3}\end{array}$$

Similarly, ratios of 60° are determined by finding the ratios of

$$\begin{array}{l}∠SQP\end{array}$$
as

$$\begin{array}{l}sin~60°\end{array}$$
=
$$\begin{array}{l}\frac{\sqrt{3}}{2}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cos~60°\end{array}$$
=
$$\begin{array}{l}\frac{1}{2}\end{array}$$

$$\begin{array}{l}tan~60°\end{array}$$
=
$$\begin{array}{l}\sqrt{3}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}cot~60°\end{array}$$
=
$$\begin{array}{l}\frac{1}{\sqrt{3}}\end{array}$$

$$\begin{array}{l}cosec~60°\end{array}$$
=
$$\begin{array}{l}\frac{2}{\sqrt{3}}\end{array}$$

$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$

$$\begin{array}{l}sec~60°\end{array}$$
=
$$\begin{array}{l}2\end{array}$$

Value of Trigonometric Ratios for Angle equal to 0 and 90 degrees

In

$$\begin{array}{l}∆ABC\end{array}$$
is a right angled triangle. If the length of side
$$\begin{array}{l}BC\end{array}$$
is continuously decreased, then value of
$$\begin{array}{l}∠A\end{array}$$
will keep on decreasing. Similarly, value of
$$\begin{array}{l}∠C\end{array}$$
is increasing as length of
$$\begin{array}{l}BC\end{array}$$
is decreasing. When BC = 0, ∠A = 0 , ∠C = 90° and AB = AC.

Taking ratios for

$$\begin{array}{l}∠A\end{array}$$
=
$$\begin{array}{l}0°\end{array}$$

$$\begin{array}{l}sin~∠A\end{array}$$
=
$$\begin{array}{l}sin~0°\end{array}$$
=
$$\begin{array}{l}\frac{BC}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{0}{AC}\end{array}$$
=
$$\begin{array}{l}0\end{array}$$

$$\begin{array}{l}cosec~0°\end{array}$$
=
$$\begin{array}{l}\frac{1}{sin~0°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{0}\end{array}$$
= Not Defined

$$\begin{array}{l}cos~∠A\end{array}$$
=
$$\begin{array}{l}cos~0°\end{array}$$
=
$$\begin{array}{l}\frac{AB}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{AC}{AC}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

$$\begin{array}{l}sec~0°\end{array}$$
=
$$\begin{array}{l}\frac{1}{cos~0°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{1}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

$$\begin{array}{l}tan~0°\end{array}$$
=
$$\begin{array}{l}\frac{sin~0°}{cos~0°}\end{array}$$
=
$$\begin{array}{l}\frac{0}{1}\end{array}$$
=
$$\begin{array}{l}0\end{array}$$

$$\begin{array}{l}cot~0°\end{array}$$
=
$$\begin{array}{l}\frac{1}{tan~0°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{0}\end{array}$$
= Not Defined

Taking ratios for

$$\begin{array}{l}∠C\end{array}$$
=
$$\begin{array}{l}90°\end{array}$$

$$\begin{array}{l}sin~∠C\end{array}$$
=
$$\begin{array}{l}sin~90°\end{array}$$
=
$$\begin{array}{l}\frac{AB}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{AC}{AC}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

$$\begin{array}{l}cosec 90°\end{array}$$
=
$$\begin{array}{l}\frac{1}{sin~90°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{1}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

$$\begin{array}{l}cos~∠C\end{array}$$
=
$$\begin{array}{l}cos~90°\end{array}$$
=
$$\begin{array}{l}\frac{BC}{AC}\end{array}$$
=
$$\begin{array}{l}\frac{0}{AC}\end{array}$$
=
$$\begin{array}{l}0\end{array}$$

$$\begin{array}{l}sec~90°\end{array}$$
=
$$\begin{array}{l}\frac{1}{cos~90°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{0}\end{array}$$
= Not Defined

$$\begin{array}{l}tan~90°\end{array}$$
=
$$\begin{array}{l}\frac{sin~90°}{cos~90°}\end{array}$$
=
$$\begin{array}{l}\frac{1}{0}\end{array}$$
= Not Defined

$$\begin{array}{l}cot~90°\end{array}$$
=
$$\begin{array}{l}\frac{1}{tan~90°}\end{array}$$
=
$$\begin{array}{l}\frac{0}{1}\end{array}$$
=
$$\begin{array}{l}0\end{array}$$

Following is the trigonometric ratios table which contains all the trigonometric ratios of standard angles:

### Solved Examples

Question 1: What is the value of tan 30+sin 60?

Solution: tan 30 = 1/√3 and sin 60 = √3/2

Adding both the values we get;

1/√3  + √3/2

Rationalising the denominator gives:

(2+√3.√3)/2√3

2+3/2√3

5/2√3

Question 2: What is the value of sin45 – cos 45?

Solution: Sin 45 = 1/√2 and cos 45 = 1/√2

Therefore, on putting the values we get:

1/√2 – 1/√2 = 0

## Video Lesson

### Trigonometric Ratios of Compound Angles

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Quiz on Trigonometric Ratios of Standard Angles