Evaluate each expression under the given conditions. tan(θ+ϕ) ; cosθ=-(1)/(3), θin Quadrant III,

step 1 So I am given the following information. So all this was given. And I need to find the tangent o

Evaluate each expression under the given conditions....

Key Concepts

Trigonometric Functions in Different Quadrants

The values and signs of trigonometric functions such as sine, cosine, and tangent vary depending on the quadrant in which the angle lies. Understanding the behavior of these functions in each of the four quadrants is crucial, as it determines whether a function’s value should be taken as positive or negative. This concept is essential for correctly evaluating trigonometric expressions that depend on quadrant-specific characteristics.

Pythagorean Identities

The Pythagorean identities, with the primary identity being sin²? + cos²? = 1, are fundamental in trigonometry. They allow the determination of one trigonometric function when another is known, by rearranging the formula to solve for the missing quantity. This method is instrumental for reconstructing full trigonometric profiles of angles from partial information, thereby enabling the complete evaluation of expressions involving multiple trigonometric functions.

Tangent Addition Formula

The tangent addition formula expresses the tangent of the sum of two angles in terms of the tangents of the individual angles: tan(? + ?) = (tan ? + tan ?) / (1 - tan ? tan ?). This identity is widely applicable in solving problems in trigonometry where the angle sum is involved, making it a key tool to evaluate the tangent function when the individual tangent values are known or can be calculated.

Transcript

00:01 So i am given the following information.

00:02 So all this was given.

00:07 And i need to find the tangent of data minus phi.

00:11 So all i know is that cosine of data is in quadrant three.

00:15 So it's somewhere here.

00:18 So the cosine is going to be negative, which is here.

00:22 And a sign of five is in quadrant five is in quadrant two.

00:27 So it's right here.

00:29 So the sign is positive, which i already have it.

00:32 But the cosine is negative.

00:34 So i need to know that's what i make, the adjustments that i need to make once i start to plug in stuff into my equation.

00:42 So the first thing i am going to do is use the subtraction formula for tangent.

00:47 So i know that this is going to become time of data minus tine of fi over 1 plus time of data, tan tangent of fi.

01:03 And now i don't know any of these values, but one thing that i do know is that the cosine of data is 1 over 3.

01:13 I can just ignore the negative.

01:14 I know it's negative because it's in quadrant 3, but to make the triangle to help me find the missing side, i don't need to put the negative.

01:23 So cosine of data is opposite, no adjacent over hypotenuse.

01:30 So i need to find this missing length.

01:32 So it's going to be 1 squared plus y square equals to 3 square.

01:37 This becomes 1 plus y squared equals to 9, minus 1, y squared is equal to 8.

01:47 I take the square root, which is going to leave y equals 2, the square root of 8, which then becomes 2 square root of 2.

01:59 This if you forgot breaks up into four times two so that's how i can simplify it to this i'm going to do a similar process for five again now i know that the sign it's opposite over hypotenuse so now i need to find this missing side so it's going to be one square plus x squared equals to four squared so this becomes comes 1 plus x squared equals to 16 minus 1 minus 1.

02:37 I have x squared equals to 15 and then i take the square root.

02:43 It leaves me with x equals radical 15.

02:50 So once i have all this information, i can label my missing side accordingly.

02:58 And that is going to help me, because i need to find the tangents, which i don't have yet...

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