Prove the following identities. sec θ = 1 cos θ \sec\theta=\frac{1}{\cos\theta} sec θ = c o s θ 1

Hey everyone. In this problem we are told that sin is equal to 16 and we are asked to find cosecant of q using trigonometric identities. We have 4 answer choices, option a, 136, option B, 56, option C, 6, and option D, 65th. So let's write down what we're trying to find, cosecant of q, And recall that cosecant of q is just equal to 1 divided by sine of q. Now we're given sine of q, so we can just substitute that in now. So cosecant of q is gonna be equal to 1 divided by 1 6th, and if we simplify this, this is just equal to 6. K. And so using our trig identity that the cosecant of q is equal to 1 divided by sine of q, we were able to find that cosecant of q is equal to 6. So the correct answer here is option c. That's it for this one. Thanks everyone for watching. See you in the next video.

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0. Functions

Trigonometric Identities

Calculus

0. Functions

Trigonometric Identities

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Problem 67

Textbook Question

Prove the following identities.
$\sec\theta=\frac{1}{\cos\theta}$

Verified step by step guidance

Start by recalling the definition of the secant function in trigonometry. The secant of an angle $ \theta $ is defined as the reciprocal of the cosine of that angle.

Express the secant function in terms of cosine: $ \sec\theta = \frac{1}{\cos\theta} $. This is the fundamental definition of the secant function.

To prove the identity $ \sec\theta = \frac{1}{\cos\theta} $, we need to show that this expression holds true for all values of $ \theta $ where $ \cos\theta \neq 0 $.

Consider the unit circle, where the cosine of an angle $ \theta $ is the x-coordinate of the point on the circle. The secant, being the reciprocal, represents the ratio of the hypotenuse to the adjacent side in a right triangle.

Since $ \sec\theta $ is defined as $ \frac{1}{\cos\theta} $, and this relationship is derived directly from the definitions of the trigonometric functions, the identity $ \sec\theta = \frac{1}{\cos\theta} $ is proven by definition.

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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 secant, relate the angles of a triangle to the ratios of its sides. The secant function, denoted as sec(θ), is defined as the reciprocal of the cosine function, specifically sec(θ) = 1/cos(θ). Understanding these functions is essential for manipulating and proving identities in trigonometry.

Reciprocal Identities

Reciprocal identities are fundamental relationships in trigonometry that express one trigonometric function in terms of another. For example, the identity sec(θ) = 1/cos(θ) illustrates how the secant function is the reciprocal of the cosine function. Mastery of these identities is crucial for simplifying expressions and proving trigonometric identities.

Proving Trigonometric Identities

Proving trigonometric identities involves demonstrating that two expressions are equivalent using algebraic manipulation and known identities. This process often requires substituting one function for another, applying reciprocal identities, and simplifying the expressions. A solid grasp of trigonometric identities and properties is necessary to effectively prove these relationships.

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