Trigonometric Identities: Overview, Questions, Preparation

Trigonometry is the study of the relation between the sides and the angles of a triangle. However, note that trigonometry holds only in the case of right-angled triangles. This article will discuss the trigonometric ratios of the angle and move to trigonometric identities and how they are derived.

Source: ncert.nic.in

sine, cosine, tangent, cosecant, secant, and cotangent are the six trigonometric ratios. 

An equation that comprises trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle(s) involved.

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

Sin2 a + cos2 a = 1

1+tan2 a = sec2 a

Cosec2 a = 1 + cot2 a

Tan θ = Sin θ/Cos θ

Cot θ = Cos θ/Sin θ

Sin (-θ) = – Sin θ

Cos (-θ) = Cos θ

Tan (-θ) = – Tan θ

Cot (-θ) = – Cot θ

Sec (-θ) = Sec θ

Cosec (-θ) = -Cosec θ

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

Let's take two angles, a and b, so,

sin (α+β)=sin(α).cos(β)+cos(α).sin(β)

sin (α–β)=sinα.cosβ–cosα.sinβ

cos (α+β)=cosα.cosβ–sinα.sinβ

cos (α–β)=cosα.cosβ+sinα.sinβ

tan (α+β)=tanα+tanβ/1–tanα.tanβ

tan (α–β)=tanα–tanβ/1+tanα.tanβ

This chapter, taught in the 10th standard, is an introduction to trigonometry.

The trigonometric ratios and the functions performed using the ratios are taught before moving to the concept of identities. It carries around 8-10 marks.

• Express the ratios cos A, tan A, and sec A in terms of sin A.

Since , cos2 A+ sin2 A = 1. Therefore,

Cos2A = 1 - sin2A i.e., cos A = 1-√sin2A

cos A =  1-sin2A

tan A = sin A /cos A = sin A / 1-sin2 A and sec A = 1/ cos A = 1/ √1 – sin 2 A

• Prove that sec A (1 – sin A) (sec A + tan A) = 1.

A (1 – sin A) (sec A + tan A) = (1/ cos A) (1 – sin A) (1 / cos A + sin A/ cos A)

= (1 – sin A) (1 + sin A) / cos2A = 1- sin2A/ cos2 A = cos2A/cos2A = 1 = RHS.

• Prove that cot A –cos AcoA +cos A = cosec A-1cosec A+1

LHS = cot A – cos A/ Cot A + cos A = (cos A/ sin A) – cos A / (cos A/sin A) + cos A

= cos A (1/sin A – 1) / cos A (1/sin A + 1) = (1/sin A -1) / (1/sin A +1) = cosec A-1 / cosec A + 1 = RHS.

Q: To which triangle does trigonometry apply?

A: Trigonometry applies only to right-angled triangles, i.e. triangles with an angle of 90 degrees.

Q: How to determine other ratios of trigonometry?

If one of the trigonometric ratios of an acute angle is known, the angle's remaining trigonometric ratios can be easily determined.

Q: What are the values of trigonometric ratios?

A: The values of trigonometric ratios at angles 0°, 30°, 45°, 60°, and 90°.

Q: What are some facts related to trigonometry?

A: The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than or equal to 1.

Q: What is the significance of the chapter?

A: This chapter is crucial for students interested in taking up engineering, architecture, etc.