Year 11 Trig: Transformations and symmetry properties

[Unkindled]

I have two questions that I've been stuck on for quite a while.

Q1. If sin(x)=0.3, cos(a)=0.6, tan(θ)=0.7, find the values of:
a) tan(π/2-θ)
b) cos(π/2+x)

Q2. Find the equation of the image of y=sin(x) for each of the following transformations:
dilation of factor 2 from the y-axis followed by dilation of factor 3 from the x-axis

If you could please explain in depth how to solve these I'd really appreciate it, I am quite familiar with symmetry properties & unit circle so question 1 should be alright but I'm not really sure why for question 2 it's "y=3sin(x/2)" not "y=3sin(2x)". I'm confused because for y-axis dilation it's always the opposite, e.g. if it's dilation of 1/2 then you write it as y=sin(2x) not y=sin(1/2x).

Thanks for your time.

 

[Dr.Peterson]

Q1. If sin(x)=0.3, cos(a)=0.6, tan(θ)=0.7, find the values of:
a) tan(π/2-θ)
b) cos(π/2+x)

... I am quite familiar with symmetry properties & unit circle so question 1 should be alright

Please show us what you have tried here, including statements of the properties you are allowed to use. Both can be done easily with angle-sum formulas, but it sounds like those aren't available to you. I need to know what is.

Q2. Find the equation of the image of y=sin(x) for each of the following transformations:
dilation of factor 2 from the y-axis followed by dilation of factor 3 from the x-axis

... I'm not really sure why for question 2 it's "y=3sin(x/2)" not "y=3sin(2x)". I'm confused because for y-axis dilation it's always the opposite, e.g. if it's dilation of 1/2 then you write it as y=sin(2x) not y=sin(1/2x).

You seem to be saying the right things, so I don't see where you are confused. A dilation by a factor of a from the y-axis corresponds to replacing x in the function with x/a, as you say ("always the opposite"), so a dilation by a factor of 2 corresponds to x/2, which is (1/2)x.

If you're asking why that is true, one way to see it is that if (u, v) satisfies the given function y = f(x), then (u/a,v) satisfies y = f(ax), since f(a*u/a) = f(u) = v. So the new graph has points 1/a as far from the axis as in the original.

 

[Unkindled]

Please show us what you have tried here, including statements of the properties you are allowed to use. Both can be done easily with angle-sum formulas, but it sounds like those aren't available to you. I need to know what is.

You seem to be saying the right things, so I don't see where you are confused. A dilation by a factor of a from the y-axis corresponds to replacing x in the function with x/a, as you say ("always the opposite"), so a dilation by a factor of 2 corresponds to x/2, which is (1/2)x.

If you're asking why that is true, one way to see it is that if (u, v) satisfies the given function y = f(x), then (u/a,v) satisfies y = f(ax), since f(a*u/a) = f(u) = v. So the new graph has points 1/a as far from the axis as in the original.

We're given the following formulae to apply:

For tan(π/2-θ), I've tried doing sin(π/2-θ)/cos(π/2-θ) so that's the same as cos(θ)/sin(θ) so I got 0.6/0.3 which is 2 but the answer is supposed to be 10/7 (1.43). As for the second one I solved it while I was waiting so I'm okay with that.

Uhh. Thank you for the explanation but I don't think I fully understand it yet. I think I understand things better with actual demonstration, so could you maybe instead of using (u, v) use a certain point to explain it for me? I would really appreciate it, thank you.

 

[Dr.Peterson]

We're given the following formulae to apply:
View attachment 13629
For tan(π/2-θ), I've tried doing sin(π/2-θ)/cos(π/2-θ) so that's the same as cos(θ)/sin(θ) so I got 0.6/0.3 which is 2 but the answer is supposed to be 10/7 (1.43). As for the second one I solved it while I was waiting so I'm okay with that.

Uhh. Thank you for the explanation but I don't think I fully understand it yet. I think I understand things better with actual demonstration, so could you maybe instead of using (u, v) use a certain point to explain it for me? I would really appreciate it, thank you.

A fact they might have listed along with these is tan(π/2 - θ) = cot(θ), which in turn is 1/tan(θ); that's why we have the name cotangent (tangent of the complement). But you can do without it.

Maybe you missed the fact that the three given trig functions are with different arguments, x, a, and θ. So for tan(π/2-θ), you aren't given cos(θ) and sin(θ), but need to use tan(θ)=0.7. As I indicated, tan(π/2 - θ) = 1/tan(θ) = 1/0.7 = 1/(7/10) = 10/7.

As for the horizontal dilation, consider g(x) = f(2x). If you know that f(6) = 5, then g(3) = f(2*3) = f(6) = 5. So the point (6, 5) on the graph of f becomes (3, 5) on the graph of g: it has been dilated (compressed or shrunk) by a factor of 1/2. You can do similarly with h(x) = f(x/2) to see that it is expanded by a factor of 2.

 

[Harry_the_cat]

We're given the following formulae to apply:
View attachment 13629
For tan(π/2-θ), I've tried doing sin(π/2-θ)/cos(π/2-θ) so that's the same as cos(θ)/sin(θ) so I got 0.6/0.3 which is 2 but the answer is supposed to be 10/7 (1.43). As for the second one I solved it while I was waiting so I'm okay with that.

Uhh. Thank you for the explanation but I don't think I fully understand it yet. I think I understand things better with actual demonstration, so could you maybe instead of using (u, v) use a certain point to explain it for me? I would really appreciate it, thank you.

Note that you are not given cos(theta) or sin(theta). You are given cos(a) and sin(x).

 

[Unkindled]

A fact they might have listed along with these is tan(π/2 - θ) = cot(θ), which in turn is 1/tan(θ); that's why we have the name cotangent (tangent of the complement). But you can do without it.

Maybe you missed the fact that the three given trig functions are with different arguments, x, a, and θ. So for tan(π/2-θ), you aren't given cos(θ) and sin(θ), but need to use tan(θ)=0.7. As I indicated, tan(π/2 - θ) = 1/tan(θ) = 1/0.7 = 1/(7/10) = 10/7.

As for the horizontal dilation, consider g(x) = f(2x). If you know that f(6) = 5, then g(3) = f(2*3) = f(6) = 5. So the point (6, 5) on the graph of f becomes (3, 5) on the graph of g: it has been dilated (compressed or shrunk) by a factor of 1/2. You can do similarly with h(x) = f(x/2) to see that it is expanded by a factor of 2.

Ah. That makes sense now. We never really learned about the other three trig functions, we just did cos, tan and sin and neither my teacher nor the textbook outlined such method so that certainly required extra knowledge. Also, arguments are certainly something to look out for. I'll need some more practice with dilation, but it's starting to make sense I think. Thank you so much for your help Dr Peterson! Do have a good rest of the day.

 

[Dr.Peterson]

There are other ways to obtain the answer; one possibility might be to use the Pythagorean identity (indirectly) to relate the tangent to the sine and cosine. Another would be to go back to the right triangle definition, and observe directly that tan(π/2 - θ) is the tangent of the other acute angle; the ratio of its opposite to adjacent is the ratio of adjacent to opposite of θ, which is the reciprocal of tan(θ).

But this is all much easier once you know a little more.