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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are fundamental in simplifying expressions and solving trigonometric equations.

Some of these important identities are discussed below:

Reciprocal Trigonometric Identities

In all trigonometric ratios, there is a reciprocal relation between a pair of ratios, which is given as follows:

• sin θ = 1/cosec θ
• cosec θ = 1/sin θ
   
• cos θ = 1/sec θ 
• sec θ = 1/cos θ
   
• tan θ = 1/cot θ
• cot θ = 1/tan θ

Pythagorean Trigonometric Identities

Pythagorean trigonometric identities are based on the Right-Triangle theorem or Pythagoras theorem, and are as follows:

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

Trigonometric Ratio Identities

As tan and cot are defined as the ratio of sin and cos, which is given by the following identities:

• tan θ = sin θ/cos θ
• cot θ = cos θ/sin θ

Trigonometric Identities of Opposite Angles

In trigonometry angle measured in the clockwise direction is measured in negative parity and all trigonometric ratios defined for negative parity of angle are defined as follows:

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

Complementary Angles Identities

Complementary angles are the pair of angles whose measure add up to 90°. Now, the trigonometric identities for complementary angles are as follows:

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

Supplementary Angles Identities

Supplementary angles are the pair of angles whose measure add up to 180°. Now, the trigonometric identities for supplementary angles are:

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• tan (180°- θ) = -tan θ
• cot (180°- θ) = -cot θ

Periodicity of Trigonometric Function

Trigonometric functions such as sin, cos, tan, cot, sec, and cosec all are periodic in nature and have different periodicity. The following identities for the trigonometric ratio explain their periodicity.

• sin (n × 360° + θ) = sin θ
• sin (2nπ + θ) = sin θ
   
• cos (n × 360° + θ) = cos θ
• cos (2nπ + θ) = cos θ
   
• tan (n × 180° + θ) = tan θ
• tan (nπ + θ) = tan θ
   
• cosec (n × 360° + θ) = cosec θ
• cosec (2nπ + θ) = cosec θ
   
• sec (n × 360° + θ) = sec θ
• sec (2nπ + θ) = sec θ
   
• cot (n × 180° + θ) = cot θ
• cot (nπ + θ) = cot θ

Where, n ∈ Z, (Z = set of all integers)

Note: sin, cos, cosec, and sec have a period of 360° or 2π radians, and for tan and cot period is 180° or π radians.

Sum and Difference Identities

Trigonometric identities for Sum and Difference of angle include the formulas such as sin(A+B), cos(A-B), tan(A+B), etc.

• 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) 

Note: Identities for sin (A+B), sin (A-B), cos (A+B), and cos (A-B) are called Ptolemy’s Identities.

Double Angle Identities

As we know, sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, and we get:

sin (θ + θ) = sinθ cosθ + cosθ sinθ

• sin 2θ = 2 sinθ cosθ

Similarly,

• cos 2θ = cos²θ – sin ²θ = 2 cos ² θ – 1 = 1 – sin ² θ
• tan 2θ = (2tanθ)/(1 – tan²θ)

Half Angle Formulas

Using double-angle formulas, half-angle formulas can be calculated. To calculate half-angle formulas replace θ with θ/2 then,

• [Tex]\sin \frac{\theta}{2}  = \pm \sqrt{\frac{1-\cos \theta}{2}}[/Tex]
• [Tex]\cos \frac{\theta}{2}  = \pm \sqrt{\frac{1+\cos \theta}{2}} [/Tex]
• [Tex]\tan \frac{\theta}{2}  = \pm\sqrt{\frac{1-\cos \theta}{1+\cos \theta}} =\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta}[/Tex]

Some more Half Angle Identities

Other than the above-mentioned identities, there are some more half-angle identities which are as follows:

• [Tex] \sin \theta=\frac{2 \tan \theta / 2}{1+\tan ^2 \theta / 2} [/Tex]
• [Tex]\cos \theta=\frac{1+\tan ^2 \theta / 2}{1- \tan ^2 \theta / 2} [/Tex]
• [Tex] \tan \theta = \frac{2 \tan \theta / 2}{1- \tan ^2 \theta / 2}[/Tex]

Product-Sum Identities

The following identities state the relationship between the sum of two trigonometric ratios with the product of two trigonometric ratios.

• [Tex]\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} [/Tex]
• [Tex]\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2} [/Tex]
• [Tex]\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2} [/Tex]
• [Tex]\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}[/Tex]

Products Identities

Product Identities are formed when we add two of the sum and difference of angle identities and are as follows:

• [Tex]\sin A \cos B=\frac{\sin (A+B)+\sin (A-B)}{2} [/Tex]
• [Tex]\cos A \cos B=\frac{\cos (A+B)+\cos (A-B)}{2} [/Tex]
• [Tex]\sin A \sin B=\frac{\cos (A-B)-\cos (A+B)}{2}[/Tex]

Triple Angle Formulas

Other than double and half angle formulas, there are identities for trigonometric ratios which are defined for triple angle. These identities are as follows:

• [Tex]\sin 3 \theta=3 \sin \theta-4 \sin ^3 \theta [/Tex]
• [Tex]\cos 3 \theta= 4 \cos^3 \theta-3 \cos \theta [/Tex]
• [Tex]\cos 3 \theta=\frac{3 \tan \theta-\tan ^3 \theta}{1-3 \tan ^2 \theta} [/Tex]

Relation between Angles and Sides of Triangle

Three rules which related the sides of triangles to the interior angles of triangles are,

• Sine Rule
• Cosine Rule
• Tangent Rule

If a triangle ABC with sides a, b, and c which are sides opposites to the ∠A, ∠B, and ∠C respectively, then

#Image

Sine Rule

Sine rule states the relationship between sides and angles of the triangle which is the ratio of side and sine of angle opposite to the side always remains the same for all the angles and sides of the triangle and is given as follows:

[Tex]\bold{\frac{\sin \angle A}{a}= \frac{\sin \angle B}{b} = \frac{\sin \angle C}{c} = k}[/Tex]

Cosine Rule 

Cosine Rule involves all the sides, and one interior angle of the triangle is given as follows:

[Tex]\bold{\cos \angle A = \frac{b^2+c^2 – a^2}{2bc}}[/Tex]

OR

[Tex]\bold{\cos \angle B = \frac{a^2+c^2 – b^2}{2ac}}[/Tex]

OR

[Tex]\bold{\cos \angle C = \frac{a^2+b^2 – c^2}{2ab}}[/Tex]

 

Tangent Rule

• Tangent Rule also states the relationship between the sides and interior angle of a triangle, using the tan trigonometric ratio, which is as follows:

• [Tex]\bold{\frac{a-b}{a+b}=\frac{\tan \left(\frac{A-B}{2}\right)}{\tan \left(\frac{A+B}{2}\right)}}[/Tex]
• [Tex]\bold{\frac{b-c}{b+c}=\frac{\tan \left(\frac{B-C}{2}\right)}{\tan \left(\frac{B+C}{2}\right)}}[/Tex]
• [Tex]\bold{\frac{c-a}{c+a}=\frac{\tan \left(\frac{C-A}{2}\right)}{\tan \left(\frac{C+A}{2}\right)}}[/Tex]

Also, Read

• Trigonometry Height and Distance
• Trigonometric Table

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant. 

• sin⁻¹(x)
• cos⁻¹(x)
• tan⁻¹(x)
• cot⁻¹(x)
• sec⁻¹(x)
• csc⁻¹(x)

Domain and Range

Domain and range of inverse trigonometric functions are: