Finding θ Without the Aid of a Calculator

[Burjam]

Homework Statement

These are three separate problems

sec θ = 2

cot θ =1

csc θ = 2

Homework Equations

I'm not sure which of the trigonometric identities to use here. Therefore, I'm not sure which equations are relevant.

The Attempt at a Solution

I've never learned how to evaluate these without a calculator. I don't even know where to start. I tried using the 1 + tan²θ = sec²θ. But I couldn't yield many results with this.

 

[berkeman]

Homework Statement

These are three separate problems

sec θ = 2

cot θ =1

csc θ = 2

Homework Equations

I'm not sure which of the trigonometric identities to use here. Therefore, I'm not sure which equations are relevant.

The Attempt at a Solution

I've never learned how to evaluate these without a calculator. I don't even know where to start. I tried using the 1 + tan^2θ = csc^2θ. But I couldn't yield many results with this.

Welcome to the PF.

Start by writing them in the form of sin, cos, tan. Those functions are defined simply in terms of them...

Then draw the triangles associated with each of them...

 

[Chestermiller]

Homework Statement

These are three separate problems

sec θ = 2

cot θ =1

csc θ = 2

Homework Equations

I'm not sure which of the trigonometric identities to use here. Therefore, I'm not sure which equations are relevant.

The Attempt at a Solution

I've never learned how to evaluate these without a calculator. I don't even know where to start. I tried using the 1 + tan^2θ = csc^2θ. But I couldn't yield many results with this.

If secθ = 2, what is cosθ?
If cotθ = 1, what is tanθ?
If cscθ = 2, what is sinθ?

Also, the equation in boldface in your posting is incorrect.

 

[Mark44]

I tried using the 1 + tan^2θ = csc^2θ.

Also, the equation in boldface in your posting is incorrect.

As well as ambiguous. To reduce the ambiguity of expressions with the square of a trig function, use parentheses, like this: sin^2(θ) or better yet, sin²(θ).

You can write exponents like this by clicking the Go Advanced button (which opens an advanced menu), and clicking the X² button.

 

[Burjam]

Welcome to the PF.

Start by writing them in the form of sin, cos, tan. Those functions are defined simply in terms of them...

Then draw the triangles associated with each of them...

The farthest I'm able to get is that

1/cos θ = 2

1/tan θ = 1

1/sin θ = 2

If secθ = 2, what is cosθ?
If cotθ = 1, what is tanθ?
If cscθ = 2, what is sinθ?

Also, the equation in boldface in your posting is incorrect.

That was a typo. I just corrected it. I was actually using 1 + tan² θ = sec² θ

In any case, I've gotten to the point where I'm at:

cosθ = 2

tanθ = 1

sinθ = 2

I'm still not sure how to proceed from here without a calculator. Normally, I would just input θ = cos^-1(2) in my calculator, but I can't do that here.

As well as ambiguous. To reduce the ambiguity of expressions with the square of a trig function, use parentheses, like this: sin^2(θ) or better yet, sin²(θ).

You can write exponents like this by clicking the Go Advanced button (which opens an advanced menu), and clicking the X² button.

My apologizes. I didn't take the time to look at the advanced menu when I posted.

 

[Mark44]

The farthest I'm able to get is that

1/cos θ = 2

1/tan θ = 1

1/sin θ = 2

These are fine.

That was a typo. I just corrected it. I was actually using 1 + tan² θ = sec² θ

In any case, I've gotten to the point where I'm at:

cos = 2

tanθ = 1

sinθ = 2

These are wrong.

From the first set of equations, if 1/cosθ = 2, then cos θ = 1/2, and similarly for the other two.

There are only a handful of angles whose cosines you are expected to know exactly (i.e., without using a calculator), but this is one of them.

I'm still not sure how to proceed from here without a calculator. Normally, I would just input θ = cos^-1(2) in my calculator, but I can't do that here.
My apologizes. I didn't take the time to look at the advanced menu when I posted.

 

[Burjam]

These are fine.
These are wrong.

From the first set of equations, if 1/cosθ = 2, then cos θ = 1/2, and similarly for the other two.

Then

cosθ = 1/2
tanθ = 1
sinθ = 1/2

There are only a handful of angles whose cosines you are expected to know exactly (i.e., without using a calculator), but this is one of them.

I still don't know exactly what it is yet. Is there a way of figuring it out or will I just have to memorize a small list of common solutions?

 

[klondike]

draw 2 right right triangles: one with 45°,45° and 90°, the other with 30°,60°,90°. Observe the sin,cos,tan. You will find out what's going on quickly.

Then

cosθ = 1/2
tanθ = 1
sinθ = 1/2



I still don't know exactly what it is yet. Is there a way of figuring it out or will I just have to memorize a small list of common solutions?

 

[Burjam]

draw 2 right right triangles: one with 45°,45° and 90°, the other with 30°,60°,90°. Observe the sin,cos,tan. You will find out what's going on quickly.

Ohhhh I see now.

a. θ = 60º
b. θ = 45º
c. θ = 30º

Wow, it's a lot simpler than I thought it would be. I'm kicking myself right now. Anyway, thanks for the help. My textbook wasn't very clear.