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.$$
\sin t=\frac{2}{3}, \cos t=\frac{\sqrt{5}}{3}
$$

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Key Concepts

Trigonometric Function Definitions

This concept involves expressing the trigonometric functions tangent, secant, cosecant, and cotangent in terms of sine and cosine. For example, tangent is defined as the ratio of sine to cosine, secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotangent is the ratio of cosine to sine. These definitions are fundamental in trigonometry and allow you to derive one trigonometric function from another.

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity (sin²t + cos²t = 1), are equations that hold for all values of the trigonometric variables. They provide a framework for transforming and simplifying trigonometric expressions, making it easier to compute and relate various functions to each other.

Rationalizing Denominators

Rationalizing denominators is an algebraic technique used to eliminate irrational numbers (like square roots) from the bottom of a fraction. This process simplifies expressions and is especially useful in expressing final answers in a standardized and more easily comparable form.

Transcript

00:01 Hi there.

00:01 So today we're working on question 4 .2, number 28, where you're asked to find tangent, co -seekent, secant, and cotangent, given sign and cosine.

00:16 Okay.

00:16 And so what this question is testing you on is the fact that sine and cosine can be used to solve for the rest of the trigonometric functions.

00:27 Okay.

00:27 And so on the right, what i've done is to define.

00:31 The trigonometric identities so that we can have them as a reference throughout our problem.

00:38 Okay? and you should have these memorized.

00:43 And so for number 28, we're dealing with specifically sine t when it equals 2 over 3 and cosine t when it equals square root 4.

01:07 5 over 3.

01:11 Okay.

01:12 And so the first function we need to solve for is tan at t.

01:22 And if we look over to our array, tan of t has two trigonometric identities.

01:28 One is one over a cotangent t or sine t over cosine t.

01:35 And since we're given sign and cosine, the second identity is probably best for this problem.

01:45 And so given that 10t of t equals sine t over cosine t, then that means that's equal to 2 over 3 over square root 5 over 3.

02:00 And that's the same as multiplying 2 over 3 times the inverse 3 over 5...

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