Table of Contents
Introduction to Trigonometric Identities
Trigonometric identities are essential equations in trigonometry that link multiple trigonometric functions of an angle. These identities are essential in solving complicated trigonometric problems, analysing triangles, and simplifying formulas. In this post, we will look at popular trigonometric identities, solve them step by step, and answer commonly asked issues with thorough answers.
Pythagorean Identities
Pythagorean identities are essential formulas in trigonometry for right triangles. They relate sides and angles and are widely used in calculations involving trigonometric functions.
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Reciprocal identities
Reciprocal identities are crucial formulas in trigonometry that establish relationships between trigonometric functions and their reciprocals. These identities are useful for simplifying expressions and solving trigonometric equation
- csc(θ) = 1 / sin(θ)
- sec(θ) = 1 / cos(θ)
- cot(θ) = 1 / tan(θ)
Quotient and cofunction identities
Quotient and cofunction identities are fundamental equations in trigonometry that connect trigonometric functions with their reciprocals and cofunctions. These identities are employed for simplifying expressions and solving trigonometric equations.
- tan(θ) = sin(θ) / cos(θ)
- cot(θ) = cos(θ) / sin(θ)
Sum and difference formulas
Sum and difference formulas are fundamental equations in trigonometry that provide relationships between the sum and difference of angles and trigonometric functions. These formulas are essential for expanding trigonometric expressions and solving trigonometric equations involving multiple angles.
- sin(α ± β) = sin(α) * cos(β) ± cos(α) * sin(β)
- cos(α ± β) = cos(α) * cos(β) ∓ sin(α) * sin(β)
- tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α) * tan(β))
Double angle formulas
Double angle formulas are fundamental equations in trigonometry that relate trigonometric functions to their double angles. These formulas are used to simplify expressions and solve problems involving trigonometric functions.
- sin(2θ) = 2 * sin(θ) * cos(θ)
- cos(2θ) = cos²(θ) – sin²(θ) or 2 * cos²(θ) – 1
- tan(2θ) = 2 * tan(θ) / (1 – tan²(θ))
Conclusions
Trigonometric identities are strong tools for simplifying trigonometric expressions, solving trigonometric equations, and comprehending trigonometric function connections. This article reviewed common trigonometric identities, offered solved examples with step-by-step instructions, and answered commonly asked problems with extensive answers. Students may flourish in trigonometry and its myriad real-world applications by knowing these identities and comprehending their applications.
Frequently asked questions on Trigonometric identities
Prove the identity: sin(θ) * cot(θ) = cos(θ).
We can use the reciprocal identity cot(θ) = cos(θ) / sin(θ). sin(θ) * cot(θ) = sin(θ) * (cos(θ) / sin(θ)) = cos(θ).
Simplify the expression: (sin(θ) + cos(θ))^2.
Expand the expression using the binomial formula. (sin(θ) + cos(θ))^2 = sin²(θ) + 2sin(θ) * cos(θ) + cos²(θ). Using the Pythagorean identity sin²(θ) + cos²(θ) = 1, the expression simplifies to 1 + 2sin(θ) * cos(θ).
Prove the identity: tan(θ) * cot(θ) = 1.
Using the quotient identity tan(θ) = sin(θ) / cos(θ) and cot(θ) = cos(θ) / sin(θ), we get: tan(θ) * cot(θ) = (sin(θ) / cos(θ)) * (cos(θ) / sin(θ)) = 1.
Find the value of cos(π/3) using the unit circle.
In the unit circle, cos(π/3) corresponds to the x-coordinate of the point (cos(π/3), sin(π/3)) which is (1/2, √3/2). Therefore, cos(π/3) = 1/2. Prove the identity: cos(θ) / sec(θ) = cos²(θ).
Prove the identity: cos(θ) / sec(θ) = cos²(θ).
Using the reciprocal identity sec(θ) = 1 / cos(θ), we get: cos(θ) / sec(θ) = cos(θ) * cos(θ) = cos²(θ).
Simplify the expression: (1 - sin²θ) / (1 + sinθ).
Using the Pythagorean identity 1 - sin²θ = cos²θ, the expression simplifies to: cos²θ / (1 + sinθ).
Prove the identity: sin(90° - θ) = cos(θ).
Using the co-function identity, sin(90° - θ) = cos(θ).
Prove the identity: sin(θ) * csc(θ) = 1.
Using the reciprocal identity csc(θ) = 1 / sin(θ), we get: sin(θ) * csc(θ) = sin(θ) * (1 / sin(θ)) = 1.
