Trigonometric Ratios and Identities

Trigonometric ratios and identities form the backbone of Trigonometry, a mathematical branch that deals with the relationships and properties of triangles and angles. Trigonometric ratios such as sine, cosine, tangent, cosecant, secant, and cotangent are Basic principles employed to delineate the connections among the angles and segments of a right-angled triangle.

1.0Trigonometric Ratios

Six trigonometric ratios are defined as sine, cosine, tangent, and their reciprocal resp as cosecant, secant, and cotangent based on the ratio of any two sides in a right-angled triangle.

Here are the key Trigonometric Ratios:

1. Sine (sin): The sine of an angle within a right triangle is established As the ratio between the length of the side opposite  to the angle and the length of the hypotenuse.

$\sin (\theta )=\frac{\text{ Opposite Side }}{\text{ Hypotenuse }}\text{ or }\sin \theta =\frac{\mathrm{P}}{\mathrm{H}}$

2. Cosine (cos): The cosine of an angle within a right triangle is defined as the ratio between the length of the adjacent side of the angle and the hypotenuse length.

$\cos (\theta )=\frac{\text{ AdjacentSide }}{\text{ Hypotenuse }}\text{ or }\cos \theta =\frac{\mathrm{B}}{\mathrm{H}}$

3. Tangent (tan): The tangent of an angle within a right triangle is defined as the ratio between the length of the opposite side of the angle and the length of the adjacent side.

$\tan (\theta )=\frac{\text{ Opposite Side }}{\text{ Adjacent Side }}\text{ or }\tan \theta =\frac{P}{B}$

4. Cosecant (cosec): The cosecant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side opposite the angle.

$\mathrm{cosec}\theta =\frac{\text{ Hypotenuse }}{\text{ Opposite side }}\text{ or }\mathrm{cosec}\theta =\frac{\mathrm{H}}{\mathrm{P}}$

5. Secant (sec): The secant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side adjacent to the angle.

$\sec (\theta )=\frac{\text{ Hypotenuse }}{\text{ Adjacent Side }}\text{ or }\sec \theta =\frac{H}{B}$

6. Cotangent (cot): The cotangent of an angle within a right triangle is defined as the ratio between the adjacent side and the length of the side opposite the angle.

$\cot (\theta )=\frac{\text{ Adjacent Side }}{\text{ Opposite Side }}\text{ or }\cot \theta =\frac{\mathrm{B}}{\mathrm{P}}$

These ratios are useful in solving various problems involving angles and distances in real-world applications such as Physics, Engineering, and Astronomy.

2.0Trigonometric Ratios and Identities Formulas

Trigonometric identities constitute equations that hold true for all values of the involved variables. They are derived from the definitions of trigonometric functions and are used extensively in simplifying trigonometric expressions and solving trigonometric equations. Some important 8 basic trigonometric identities include: 

• Pythagorean Identities.
• Reciprocal Identities.
• Opposite Angle Identities
• Complementary Angle Identities
• Supplementary Angle Identities
• Product Identities
• Sum and Difference Identities
• Sum to Product Identities
• Double Angle Identities
• Triple Angle Identities
• Half Angle Identities 

Pythagorean Identities:

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

Understanding and applying these trigonometric ratios and identities are essential skills for students and professionals in fields related to mathematics and its applications.

Reciprocal Identities

• $\sin \theta =\frac{1}{\mathrm{cosec}\theta }$
• $\cos \theta =\frac{1}{\sec \theta }$
• $\tan \theta =\frac{1}{\cot \theta }$
• $\cot \theta =\frac{1}{\tan \theta }$
• $\sec \theta =\frac{1}{\cos \theta }$
• $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$

Opposite Angle Identities.

• 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

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

Supplementary Angles Identities

• sin (180 + θ) = sin θ
• cos (180 + θ) = – cos θ
• tan (180 + θ) = – tan θ
• cot (180 + θ) = – cot θ
• sec (180 + θ) = – sec θ
• cosec (180 + θ) = cosec θ

Sum and Difference Identities:

• 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
• $\tan (\mathrm{A}+\mathrm{B})=\frac{\tan \mathrm{A}+\tan \mathrm{B}}{1-\tan \mathrm{A}\cdot \tan \mathrm{B}}$
• $\tan (\mathrm{A}-\mathrm{B})=\frac{\tan \mathrm{A}-\tan \mathrm{B}}{1+\tan \mathrm{A}\cdot \tan \mathrm{B}}$

Sum to Product Identities

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

Double - Angle Identities:

• $\sin (2\theta )=2\sin \theta \cdot \cos \theta =\frac{2\tan \theta }{1+{\tan }^2\theta }$
• $\cos (2\theta )={\cos }^2\theta -{\sin }^2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$

= 2cos²θ – 1

= 1 – 2 sin²θ

• $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$
• $\sec 2\theta =\frac{{\sec }^2\theta }{2-{\sec }^2\theta }$
• $\mathrm{cosec}2\theta =\frac{(\sec \theta \cdot \mathrm{cosec}\theta )}{2}$
• 1 + cos 2θ = 2 cos² θ
• 1 – cos 2θ = 2 sin² θ

Triple Angle Identities

• sin 3θ = 3 sinθ – 4 sin³θ
• cos 3θ = 4 cos³θ – 3 cosθ
• $\tan 3\theta =\frac{3\tan \theta \cdot {\tan }^3\theta }{1-3{\tan }^2\theta }$

Half angle Identities

• $\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}}$
• $\cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}$
• $\tan \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}$

$=\frac{1-\cos \theta }{\sin \theta }$

$=\frac{\sin \theta }{1+\cos \theta }$

• $\cot \frac{\theta }{2}=\frac{1+\cos \theta }{\sin \theta }$
• ${\tan }^2\frac{\theta }{2}=\frac{1-\cos \theta }{1+\cos \theta }$
• ${\cot }^2\frac{\theta }{2}=\frac{1+\cos \theta }{1-\cos \theta }$

3.0Trigonometric Table

Below Trigonometric Table is given:

4.0Questions on Trigonometric Ratios and Identities

Example 1: If sin θ + sin² θ = 1, then the value of cos²θ + cos⁴θ  is equal to :

(A) 0 (B) 5 (C) $\frac{1}{2}$ (D) 1 

Ans.  (D)

Solution: Given sin θ + sin² θ = 1 

Using formula sin² θ + cos² θ = 1

⇒ cos² θ = 1 – sin² θ 

sin θ + sin² θ = 1 … (1)

⇒ sin θ = 1 – sin² θ = cos² θ 

⇒ sin θ = cos² θ 

Putting in given equation (1)

⇒ cos² θ + cos⁴ θ  = 1

Example 2: 4(sin⁶ θ + cos⁶ θ) – 6 (sin⁴ θ + cos⁴ θ) is equal to

(A) 0 (B) 1 (C) –2 (D) None of these

Ans. (C)

 Solution: 4[(Sin² θ + cos² θ)³ – 3 sin² θ cos² θ (sin² θ + cos² θ)] –6 [(sin² θ + cos² θ)² – 2sin ² θ cos²θ]

   = 4 [1–3sin ² θ cos² θ] –6[1–2sin² θ cos²θ]

  = 4–12sin² θ cos ² θ –6 + 12 sin² θ cos² θ = –2

  Option (c ) is correct.

Example 3: Prove that $\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\mathrm{cosec}\theta -\cot \theta$

Solution:

$\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\sqrt{\frac{(1-\cos \theta )(1-\cos \theta )}{(1+\cos \theta )(1-\cos \theta )}}=\frac{1-\cos \theta }{\sin \theta }=\mathrm{cosec}\theta -\cot \theta$

Example 4:  $\frac{\sin 7\theta -\sin 5\theta }{\cos 7\theta +\cos 5\theta }$

Solution: Using formula 

$=\frac{2\cos \left(\frac{7\theta +5\theta }{2}\right)\sin \left(\frac{7\theta -5\theta }{2}\right)}{2\cos \left(\frac{7\theta +5\theta }{2}\right)\cos \left(\frac{7\theta -5\theta }{2}\right)}$

$=\frac{2\cos 6\theta \cdot \sin \theta }{2\cos 6\theta \cdot \cos \theta }=\tan \theta$

Example 5: $\frac{\sin \theta +\sin 3\theta +\sin 5\theta }{\cos \theta +\cos 3\theta +\cos 5\theta }$

Solution = $\frac{(\sin \theta +\sin 5\theta )+\sin 3\theta }{(\cos \theta +\sin 5\theta )+\cos 3\theta }$

$=\frac{2\sin \left(\frac{\theta +5\theta }{2}\right)\cos \left(\frac{\theta -5\theta }{2}\right)+\sin 3\theta }{2\cos \left(\frac{0+5\theta }{2}\right)\cos \left(\frac{\theta -5\theta }{2}\right)+\cos 3\theta }$

$=\frac{2\sin 3\theta \cdot \cos (-2\theta )+{\sin }^3\theta }{2\cos 3\theta \cos (-2\theta )+{\cos }^3\theta }=\frac{\sin 3\theta [2\cos 2\theta +1]}{\cos 3\theta [2\cos 2\theta +1]}$

$=\frac{\sin 3\theta }{\cos 3\theta }=\tan 3\theta$

5.0Sample Question from Trigonometric Ratios and Identities

1. What are the Pythagorean identities?

Ans: The Pythagorean identities are a set of three identities derived from the Pythagorean theorem. They are:

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