Evaluate each expression under the given conditions. sin(θ-ϕ) ; tanθ=(4)/(3), θin Quadrant III,

step 1 So I have to find out what is the sign of data minus phi. And I already know that tangent of dat

Evaluate each expression under the given conditions....

Key Concepts

Sine Difference Identity

The sine difference identity states that the sine of the difference between two angles equals the sine of the first angle times the cosine of the second minus the cosine of the first angle times the sine of the second. This formula is essential for breaking down and evaluating expressions that involve the subtraction of angles.

Reference Triangle Construction

Reference triangle construction involves creating a right triangle based on a given trigonometric ratio. This method helps in determining other trigonometric values of the angle by using the relationships among the sides, allowing one to recover sine, cosine, or other ratios from a known value.

Quadrant Analysis

Quadrant analysis is used to determine the sign (positive or negative) of trigonometric functions based on the angle's location in the coordinate plane. Knowing the quadrant in which an angle resides is crucial, as each quadrant has its own sign rules for sine, cosine, and tangent.

Pythagorean Identity

The Pythagorean identity, expressed as the sum of the squares of sine and cosine equaling one, is a fundamental result in trigonometry. It is used to find an unknown trigonometric value when one value is known, ensuring consistency and correctness in the evaluation of expressions.

Transcript

00:01 So i have to find out what is the sign of data minus phi.

00:05 And i already know that tangent of data is four thirds.

00:11 And i also know that it's in quadrant, um, beta data is in quadrant three.

00:18 So put this on the side.

00:20 And i also know that sign of data, no sign of phi, is equal to radical 10, negative radical 10.

00:34 Over 10 and phi is in quadrant 4.

00:42 So i am going to use the subtraction formula for signs to rewrite this.

00:48 So this is the same thing as sine data cosine pi minus sine of 5 cosine of data.

01:01 And now notice that the only information that i do have is sign of 5, with this information given.

01:10 So if i know that tangent of data is 4 over 3, i go back to drawing the right triangle.

01:20 And if this is data, then tangent is opposite over adjacent.

01:27 So that makes the hypotenuse 5, because 3 square plus 4 square gives you 5 square.

01:33 And then i have to do the same, something similar.

01:38 For 5.

01:40 I know sign is the same thing as opposite.

01:44 So this is radical 10 over hypotenuse which is 10 so that i'm going to use that to find this missing side.

01:51 And again, i am assuming that this is a right triangle.

01:55 So then i am looking for this missing length plus radical 10 square equals to 10 square.

02:06 So this is x squared plus this is just 10 by itself equals to 100.

02:14 This is minus 10, minus 10, x squared equals to 90...

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