In Exercises 25–28, sin t and cos t are given. Use...

In Exercises 25–28, sin t and cos t are given. Use identities to find tan t, csc t, sec t, and cot t. Where necessary, rationalize denominators.
$$
\sin t=\frac{1}{3}, \cos t=\frac{2 \sqrt{2}}{3}
$$

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a process used in simplifying expressions that contain irrational numbers in the denominator. This step is often required to express the final results in a more standardized and acceptable form, ensuring that no radicals remain in the denominator.

Reciprocal Identities

Reciprocal identities define the relationship between the primary trigonometric functions and their reciprocals. Cosecant is the reciprocal of sine, and secant is the reciprocal of cosine. Utilizing these definitions enables direct computation of csc t and sec t from sin t and cos t, respectively.

Trigonometric Identities

Trigonometric identities are equations that express relationships between the trigonometric functions and can be used to derive one function from another. In problems where some function values are given, these identities help find related functions by using relationships such as the quotient identity, reciprocal identity, and Pythagorean identity.

Quotient Identities

The quotient identities relate the tangent and cotangent functions to sine and cosine. Specifically, tan t is defined as sin t divided by cos t, and cot t as cos t divided by sin t. These identities allow the extraction of a ratio between sine and cosine values when they are known.

Transcript

Please give Ace some feedback