# Trigonometric Ratios

Trigonometric ratios are the ratios of the side lengths of a right-angled triangle. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). There are three more trigonometric ratios, cosecant (cosec), secant (sec), and cotangent (cot), that is, the inverse of sine, cosine, and tangent.

Thus, it is used to determine the ratios of any 2 sides of a right-angled triangle with respect to specific angles.

The value of these trigonometric ratios is calculated using the measure of any special acute angle Î¸ in a right-angled triangle.

## Formulas

The standard formulas for the given trigonometric ratios for Î¸ (where Î¸ is an acute angle) are:

Sine (sin):  It is the ratio of the opposite side (perpendicular side) to Î¸ to the hypotenuse

âˆ´ sin Î¸ = Opposite side/ Hypotenuse

Cosine (cos): It is the ratio of the adjacent side to Î¸ to the hypotenuse

âˆ´ cos Î¸ = Adjacent side/ Hypotenuse

Tangent (tan): It is the ratio of the opposite side to Î¸ to the side adjacent to Î¸

âˆ´ tan Î¸ = Opposite side/ Adjacent side

Cosecant (cosec): It is the multiplicative inverse of sine

âˆ´ cosec Î¸ = 1/sin Î¸ = Hypotenuse /Opposite side

Secant (sec): It is the multiplicative inverse of cosine

âˆ´ sec Î¸ = 1/cos Î¸ = Hypotenuse/ Adjacent side

Cotangent (cot): It is the multiplicative inverse of tangent

âˆ´ cot Î¸ = 1/tan Î¸ = Adjacent side/Opposite side

In the given right-triangle ABC, the trigonometric ratios are shown:

Thus, with respect to âˆ C, the ratios of trigonometry are:

• sine C = Side opposite to âˆ C/Hypotenuse = AB/AC
• cos C = Side adjacent to âˆ C/Hypotenuse = BC/AC
• tan C = Side opposite to âˆ C/Side adjacent to âˆ ) = AB/BC = sin âˆ C/cos âˆ C
• cosec C= 1/sin C = Hypotenuse/ Side Opposite to âˆ C = AC/AB
• sec C = 1/cos C = Hypotenuse/ Side Opposite to âˆ C = AC/BC
• cot C = 1/tan C = Side adjacent to âˆ C/Side opposite to âˆ C = BC/AB

All the ratios written above for âˆ C in all possible right triangles will be the same. The same being with âˆ A

## An Easy Way to Remember Trigonometric Ratios

The word sohcahtoa can be used to remember the trigonometric ratio formulas of sine, cosine, and tangent:

## Trigonometric Ratios Table

The trigonometric ratios table displays the values of trigonometric ratios for the standard angles 0Â°, 30Â°, 45Â°, 60Â°, and 90Â° used in several trigonometric calculations.

## Trigonometric Ratios Identities

Several trigonometric ratios identities make our calculations simpler such as:

• sin2 Î¸ + cos2 Î¸ = 1
• 1 + tan2 Î¸ = sec2 Î¸
• 1 + cot2 Î¸ = cosec2 Î¸

There are also some variations of the above 3 identities, which are nothing but rearranging the ones given above.

### Trigonometric Ratios of Complementary Angles Identities

The complement of an angle Î¸ is (90Â° – Î¸). Accordingly, the trigonometric ratios of complementary angles are:

• sin (90Â°- Î¸) = cos Î¸
• cos (90Â°- Î¸) = sin Î¸
• cosec (90Â°- Î¸) = sec Î¸
• sec (90Â°- Î¸) = cosec Î¸
• tan (90Â°- Î¸) = cot Î¸
• cot (90Â°- Î¸) = tan Î¸

### Other Trigonometric Ratios Identities

The sum, difference, and product trigonometric ratios identities are:

• sin (A + B) = sin A cos B + cos A sin B
• sin (A – B) = sin A cos B – cos A sin B
• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)
• tan (A – B) = (tan A – tan B)/ (1 + tan A tan B)
• cot (A + B) = (cot A cot B – 1)/(cot B – cot A)
• cot (A – B) = (cot A cot B + 1)/(cot B – cot A)
• 2 sin Aâ‹…cos B = sin(A + B) + sin(A – B)
• 2 cos Aâ‹…cos B = cos(A + B) + cos(A – B)
• 2 sin Aâ‹…sin B = cos(A – B) – cos(A + B)

The half, double, and triple-angles trigonometric ratios identities

• sin 2Î¸ = 2 sinÎ¸ cosÎ¸
• cos 2Î¸ = cos2Î¸ – sin2Î¸ = = 2 cos2Î¸ â€“ 1 = 1 – 2 sin2Î¸ = (1 – tan2 Î¸)/(1 + tan2 Î¸)
• sec 2Î¸ = sec2 Î¸/(2-sec2 Î¸)
• cosec 2Î¸ = (sec Î¸. cosec Î¸)/2
• cot 2Î¸ = (cot Î¸ – tan Î¸)/2
• sin 3Î¸ = 3sin Î¸ – 4sin3Î¸
• cos 3Î¸ = 4cos3Î¸ – 3cos Î¸
• tan 3Î¸ = (3tanÎ¸ – tan3Î¸)/(1 – 3tan2Î¸)

Find the value of tan Î¸ if sin Î¸ = 14/5 and cos Î¸ =6/9

Solution:

As we know,
tan Î¸ = sin Î¸/cos Î¸, here sin Î¸ = 14/5 and cos Î¸ = 6/9
= (14/5)(6/9)
= 63/15