Verify Trigonometric Identity

[harpazo]

Verify csc(pi/2 + s) = csc(pi/2 - s).

Should I convert cosecant to sine and then use the addition formulas for sines?

In other words, use sin(s + t) = sin s cos t + cos s sin t on the side and sin(s - t) = sin s cos t - cos s sin t on the right side.

 

[Dr.Peterson]

That's certainly one way.

I avoid "should", because there are often alternatives that are better, if one thinks of them. Verifying identities is an opportunity for creativity!

 

[harpazo]

That's certainly one way.

I avoid "should", because there are often alternatives that are better, if one thinks of them. Verifying identities is an opportunity for creativity!

I love proving trig identities. I am going to press on using the addition formulas and post my work if I get stuck somewhere along the way.

 

[Otis]

… Should I convert cosecant to sine and then use the [addition/subtraction] formulas for [sine]?

I think that's a good idea. After you write it out, two simple steps finish the proof.

?

 

[MarkFL]

Imagine two points on the unit circle both initially at \((0,1)\). Both points begin moving at the same speed, where one point moves in a counter clockwise direction and the other in a clockwise direction. Can you see that the \(y\)-coordinate of both points is always the same, and how this correlates to the identity you've been given to prove? If you can, please explain how you see this correlation.

Of course we should be able to prove these identities algebraically, but there's nothing wrong with visualization to bolster our understanding.

 

[MarkFL]

You can also use the identity:

[MATH]\sin(\pi-\theta)=\sin(\theta)[/MATH]
to easily prove this identity. You may recall I explained a visualization for this identity, complete with a live graph.

 

[harpazo]

You can also use the identity:

[MATH]\sin(\pi-\theta)=\sin(\theta)[/MATH]
to easily prove this identity. You may recall I explained a visualization for this identity, complete with a live graph.

Yes, you did provide a live graph for me to see what's going on. I like visualization to learn but not when compared to algebra. There's something about working out the algebra that makes me feel complete.

 

[Otis]

… I like visualization to learn but not when compared to algebra …

I'm not entirely sure what you mean, harpazo, so I'll just comment that I often find myself recalling various visualizations (eg: illustrations, animations, color-coded diagrams), as I work through exercises with paper and pencil. They can provide both context and confidence (that I'm on the right track). Sometimes, it's a visualization in the background that lends insight or deeper meaning to what I'm thinking about; the symbolic expressions and equations I've written on paper become "alive" for me (so to speak), as I work through algebraic steps.

I think it's good to keep as many tools in the toolbox as possible. The more viewpoints one employs, the closer to the 'big picture' they get.

?

 

[harpazo]

I'm not entirely sure what you mean, harpazo, so I'll just comment that I often find myself recalling various visualizations (eg: illustrations, animations, color-coded diagrams), as I work through exercises with paper and pencil. They can provide both context and confidence (that I'm on the right track). Sometimes, it's a visualization in the background that lends insight or deeper meaning to what I'm thinking about; the symbolic expressions and equations I've written on paper become "alive" for me (so to speak), as I work through algebraic steps.

I think it's good to keep as many tools in the toolbox as possible. The more viewpoints one employs, the closer to the 'big picture' they get.

?

No question about it. Right now, I am facing a dark storm that came upon me via COVID-19. Got placed on furlough without pay with the hope of returning when the pandemic is contain. This right now takes preference over math, easy or not. I cannot do math not knowing what the next few weeks will be like.