Prove the following identities.tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan2\theta=\frac{2\tan\theta}{1-\$$tan^2$$\theta}tan2θ=1−tan2θ2tanθ

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 1 - Functions

문제 87

1장, 문제 87

Prove the following identities.
$\tan\)2$\theta$=\(\frac{2\tan\theta}{1-\tan^2\theta}$

검증된 단계별 안내

Start by recalling the double angle identity for tangent: $ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} $. This is a standard trigonometric identity.

To prove this identity, let's use the sine and cosine double angle identities: $ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $ and $ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) $.

Express $ \tan(2\theta) $ in terms of sine and cosine: $ \tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} $. Substitute the double angle formulas: $ \tan(2\theta) = \frac{2\sin(\theta)\cos(\theta)}{\cos^2(\theta) - \sin^2(\theta)} $.

Now, express $ \tan(\theta) $ as $ \frac{\sin(\theta)}{\cos(\theta)} $. Substitute this into the expression: $ \tan(2\theta) = \frac{2\left(\frac{\sin(\theta)}{\cos(\theta)}\right)}{1 - \left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2} $.

Simplify the expression: $ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} $. This matches the given identity, thus proving it.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, reciprocal identities, and angle sum/difference identities, which provide relationships between different trigonometric functions.

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties and behavior of the tangent function is crucial for manipulating and proving identities involving tangent.

Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles (2θ) in terms of single angles (θ). For example, the double angle formula for tangent is tan(2θ) = 2tan(θ)/(1 - tan²(θ)). These formulas are essential for simplifying expressions and proving identities, particularly in calculus and higher-level mathematics.

Prove the following identities.tan2θ=2tanθ1−tan2θ\(\tan\)2\(\theta\)=\(\frac{2\tan\theta}{1-\tan^2\theta}\)tan2θ=1−tan2θ2tanθ

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Trigonometric Identities

Tangent Function

Double Angle Formulas

Prove the following identities.
$\tan\)2\(\theta\)=\(\frac{2\tan\theta}{1-\tan^2\theta}$