Prove the following identities. tan θ = sin θ cos θ \tan\theta=\frac{\sin\theta}{\cos\theta} tan θ = c o s θ s i n θ

Hey everyone. In this problem we are told that sine of p is equal to 16 and cosine of p is equal to the square root of 35 divided by 6. We are asked to find the cotangent of p using trigonometric identities. We have 4 answer choices. Option a is that it is equal to 6, option b, 1 divided by the square root of 35, option c, the square root of 35, and option d, 6 divided by the square root of 35. So let's start with what we're looking for. We're looking for the cotangent of p, and we want to write this in a way where we use sine and cosine, because we know what those are. Now recall that the cotangent is equivalent to 1 divided by the tangent. Okay, so cotangent of p is equal to 1 divided by tan p. And we know that we can write the tangent as sine divided by cosine. So we can write our cotangent of p as 1 divided by sine of p divided by cosine of p. Using our reciprocal rule to do this division, we can write cotangent of p as cosine of p divided by sine of p, and we know both of those values, so we can go ahead and substitute those in, and this is gonna be equal to the square root of 35 divided by 6, all divided by 16. This is just equal to the square root of 35 divided by 6 multiplied by 6 divided by 1. Okay. And this is going to give us just the square root of 35. Okay, those sixes are going to divide it. And so what we found using trig identities, is that the cotangent of p is going to be equal to the square root of 35. Comparing this to our answer choices, we can see that the correct answer is option c. Thanks, everyone, for watching. I hope this video helped. See you in the next one.

0. Functions

Trigonometric Identities

Calculus

0. Functions

Trigonometric Identities

2:26 minutes

Problem 68

Textbook Question

Prove the following identities.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Verified step by step guidance

Start by recalling the definition of the tangent function in terms of sine and cosine: $ \tan\theta = \frac{\sin\theta}{\cos\theta} $. This is a fundamental trigonometric identity.

To prove the identity $ \tan\theta = \frac{\sin\theta}{\cos\theta} $, we need to express $ \tan\theta $ in terms of $ \sin\theta $ and $ \cos\theta $.

The tangent of an angle $ \theta $ in a right triangle is defined as the ratio of the opposite side to the adjacent side. In terms of the unit circle, this translates to $ \tan\theta = \frac{y}{x} $, where $ y = \sin\theta $ and $ x = \cos\theta $.

Thus, substituting the unit circle definitions, we have $ \tan\theta = \frac{\sin\theta}{\cos\theta} $, which matches the given identity.

Therefore, the identity $ \tan\theta = \frac{\sin\theta}{\cos\theta} $ is proven using the definitions of sine, cosine, and tangent in the context of the unit circle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. The tangent function, specifically, is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. Understanding these functions is crucial for proving identities and solving problems in trigonometry.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The identity an heta = rac{ ext{sin} heta}{ ext{cos} heta} is a fundamental identity that expresses tangent in terms of sine and cosine. Familiarity with these identities is essential for simplifying expressions and proving relationships between different trigonometric functions.

Proof Techniques

Proof techniques in mathematics involve logical reasoning to demonstrate the truth of a statement or identity. Common methods include direct proof, proof by contradiction, and using known identities to derive new results. In the context of trigonometric identities, one often uses algebraic manipulation and substitution to establish the validity of the identity being proved.

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Prove the following identities.tanθ=sinθcosθ\tan\theta=\frac{\sin\theta}{\cos\theta}tanθ=cosθsinθ

Key Concepts

Trigonometric Functions

Trigonometric Identities

Proof Techniques

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Prove the following identities.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$