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 θ in a right-angled triangle.

Formulas

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

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

∴ sin θ = Opposite side/ Hypotenuse

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

∴ cos θ = Adjacent side/ Hypotenuse

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

∴ tan θ = Opposite side/ Adjacent side

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

∴ cosec θ = 1/sin θ = Hypotenuse /Opposite side

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

∴ sec θ = 1/cos θ = Hypotenuse/ Adjacent side

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

∴ cot θ = 1/tan θ = Adjacent side/Opposite side

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

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

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

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

Acronyms	Descriptions	Formulas
SOH	Sine is Opposite over Hypotenuse	Sin θ = Opposite side/ Hypotenuse
CAH	Cosine is Adjacent over Hypotenuse	Cos θ = Adjacent side/ Hypotenuse
TOA	Tangent is Opposite over Adjacent	Tan θ = Opposite side/ Adjacent side

Trigonometric Ratios Table

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

Trigonometric Ratios Identities

Several trigonometric ratios identities make our calculations simpler such as:

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• 1 + cot² θ = cosec² θ

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 θ is (90° – θ). Accordingly, the trigonometric ratios of complementary angles are:

• sin (90°- θ) = cos θ
• cos (90°- θ) = sin θ
• cosec (90°- θ) = sec θ
• sec (90°- θ) = cosec θ
• tan (90°- θ) = cot θ
• cot (90°- θ) = tan θ

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 A⋅cos B = sin(A + B) + sin(A – B)
• 2 cos A⋅cos B = cos(A + B) + cos(A – B)
• 2 sin A⋅sin B = cos(A – B) – cos(A + B)

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

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos²θ – sin²θ = = 2 cos²θ – 1 = 1 – 2 sin²θ = (1 – tan² θ)/(1 + tan² θ)
• sec 2θ = sec² θ/(2-sec² θ)
• cosec 2θ = (sec θ. cosec θ)/2
• cot 2θ = (cot θ – tan θ)/2
• sin 3θ = 3sin θ – 4sin³θ
• cos 3θ = 4cos³θ – 3cos θ
• tan 3θ = (3tanθ – tan³θ)/(1 – 3tan²θ)