If cost=-(1)/(3), and t is in quadrant III, find sint, sect, csct, tant, cott.

Name: Problem 39, Chapter 5: Precalculus
Uploaded: 2023-08-04T01:12:30-08:00
Duration: 7 min 16 s
Channel: Willis James
Description: If cost=-(1)/(3), and t is in quadrant III, find sint, sect, csct, tant, cott.

step 1 Hello, welcome to this lesson. In this lesson we have cos t that's equal to negative one on thre

If cost=-(1)/(3), and t is in quadrant III, find sint, sect,...

Key Concepts

Pythagorean Identity

This concept allows you to determine one trigonometric function from another by using the relationship sin²t + cos²t = 1. In problems where one of these functions is known, you can solve for the other by isolating the desired term, keeping in mind to take the square root, which leads to a positive or negative value based on the angle's quadrant.

Quadrant Sign Conventions

Knowing the quadrant in which an angle lies is crucial because it determines the signs of the trigonometric functions. For instance, in Quadrant III both sine and cosine are negative, which affects all derived functions such as tangent (sin/cos) and their reciprocals.

Reciprocal Trigonometric Functions

This concept relates functions such as secant and cosecant to cosine and sine respectively by the relationships sec t = 1/cos t and csc t = 1/sin t. These definitions are useful for converting between different trigonometric expressions and solving problems where reciprocal functions are required.

Tangent and Cotangent Definitions

Tangent and cotangent are defined in terms of sine and cosine as tan t = sin t/cos t and cot t = cos t/sin t. These definitions help in calculating the tangent and cotangent of an angle directly from sine and cosine values, providing a straightforward way to transition among various trigonometric functions.

Transcript

00:01 Hello, welcome to this lesson.

00:02 In this lesson we have cos t that's equal to negative one on three and we are looking for the other the five other trigonometric functions.

00:15 So here we have the the angle t that is in the third quadrant.

00:21 We have the first, the first, the second, the third.

00:26 So that's an acronym that is used we call it cast.

00:30 The first quadrant we have all of the trigonometric functions that are positive in the second quadrant.

00:40 We have sine that is positive.

00:42 We have the the third quadrant tan is positive.

00:47 Alright so if tan is positive then sine and cos negative.

00:58 Okay so here cot would also be positive.

01:01 Alright and now we'd have to use the triangle, the right angle triangle to actually find the other five.

01:11 Okay the other five trigonometric functions.

01:14 So let me put the angle t here and cos t is equal to the side that is adjacent all over the hypotenuse.

01:27 Alright so we would have the one here and a three over there.

01:36 Then from pythagoras we have the hypotenuse or the longest side squared that is equal to the sum of squares of the other two sides.

01:51 So we have a b squared plus b c squared.

01:59 So 3 squared is equals to 1 squared plus b c squared and this is 9 equals to 1 plus b c squared.

02:15 So here b c squared now becomes 9 minus 1 which is equals to 8.

02:24 Alright so we have b c that is equals to square root of 8 which is equals to square root of 4 times square root of 2 and that is equal to 2 square root of 2.

02:40 Alright so now we have this side which is the opposite as 2 square root of 2.

02:54 Alright and now let's proceed to find the other trigonometric functions.

03:03 So sine t this will be equals to the side that is opposite over the hypotenuse.

03:13 Alright so this becomes 2 square root of 2 on the third and here we have sine that is also negative.

03:26 Alright of course we are in the third quadrant.

03:30 Alright so this is what the sine t becomes.

03:35 Now let's move to sec t...

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