Please help on this problem

[dxs]

When it comes to exact values, I have real struggles, and was wondering if someone could guide me through the solving steps of this question.

 

[Otis]

Hi dxs. One approach is to use right-triangle trig definitions.

sin(x) = opposite/hypotenuse
tan(x) = opposite/adjacent

Do those look familiar?

csc(x) is the reciprocal of sin(x), so what would be the right-triangle definition for csc(x)?

Next, draw a representative right-triangle and use the given ratio -4/sqrt(7) to label the opposite side and hypotenuse. An hypotenuse is not negative, so mind where the negative sign goes.

Use the Pythagorean Theorem, to find the adjacent side. Remember, cos(theta) must be positive, in this exercise, so choose the correct sign.

You'll then have the numbers you need, to write tan(theta) as opposite/adjacent.

Questions?

?

 

[dxs]

Hi dxs. One approach is to use right-triangle trig definitions.

sin(x) = opposite/hypotenuse
tan(x) = opposite/adjacent

Do those look familiar?

csc(x) is the reciprocal of sin(x), so what would be the right-triangle definition for csc(x)?

Next, draw a representative right-triangle and use the given ratio -4/sqrt(7) to label the opposite side and hypotenuse. An hypotenuse is not negative, so mind where the negative sign goes.

Use the Pythagorean Theorem, to find the adjacent side. Remember, cos(theta) must be positive, in this exercise, so choose the correct sign.

You'll then have the numbers you need, to write tan(theta) as opposite/adjacent.

Questions?

?

In my notes I have that csc(x) = r/y but im not sure where to draw the triangle from there.

 

[Otis]

Okay, you're using symbol r to represent the hypotenuse, and symbols y and x represent the opposite side and the adjacent side (respectively). If you're not sure what I mean by 'opposite' and 'adjacent', then let us know.

You're correct: csc(x) = 1/sin(x) = r/y.

We're given that csc(theta) is -4/√7.

Therefore, its reciprocal sin(theta) is (-√7)/4.

Now, draw a right triangle having an acute angle theta, and label the hypotenuse (r) as having length 4. The side opposite angle theta (y) has directed length -√7. Label it.

Can you find the third side length (the side adjacent to angle theta)?

?

 

[dxs]

Would the exact value of tan(x) = (-√7)/3 ?

 

[Otis]

Yes! Good job.

In my previous statements, I was using variable x to mean any angle (in general).

I'd used symbol theta to represent the specific (fixed) angle in this exercise.

Therefore, I'd say tan(theta) = (-√7)/3.

?