Please help me, I am completely lost with this

Question

Using the following values, how many other trigonometric values of angles between 0 and 180 degrees can you find (without using a calculator)? Don't forget about complementary angles, co-terminal angles and other trig rules. Show work and justification for each value.

sin(5°)=0.0872; cos(45°)=0.7071; sin(60°)=0.8660; cos(60°)=0.5

You need to produce at least 10 different functions and use at least 5 different identities. Show your work and use reasoning.

Answer 1

Not sure I understand the question in total. But let me take a shot at something. Knowing the sign of a 5° angle you can also identify the following angles between 0 and 180°: 5°, 90-5°, 180-5°, 180-10°. So you can start out with the given which is the sign of 5° equal to 0.0872. Then I used the trig identity of the difference between angles: sin (90-5) = sin 90 cos5-cos 90sin5, which turned out to be 1xcos5 and from Pathag cos 5 = 0.996 so sin 85 = .996.

You can then repeat this using the identity for cosine (90-5) and then the tangent (90-5).

Then using the reference angle in the second quarter we automatically know trig functions for 180-5 by applying the identity for the difference you get three more trig values for angle 175°.

This can then be repeated for the trig functions of 170° by taking the relationship of (180-10°). Here we need to use the double angle formula to find the trig functions for 10° in terms of the trig functions for 5°. For example the sin 10° = 2 sin 5cos5 and the cosine 10 = 1 -2sin²5.

So you have one great deal of algebra to do.

Answer 2

The unit circle's radius is one, hence the name unit circle. Unit circle triangles have hypotenuse of length 1.

For points on the unit circle, the sin function essentially gives you the point's y value (opposite = y, hypotenuse = 1) while cos essentially gives you the point's x value (adjacent = x, hypotenuse = 1).

sin(theta) = cos(90-theta)

This should give you at least 4 angles at the start.

Since 0 degrees points in the x direction and 90 degrees points in the y direction, we are essentially flipping x and y and changing the direction from counterclockwise to clockwise. Since sin gives the y value and cos gives the x value, we are essentially mirroring them.

tan(theta) = sin(theta) / cos(theta)

Opposite over adjacent. This is y/x.

You can use this one for theta = 30, 45, and 60 at the very least.

sin(theta) = sin(180-theta) = sin(180 + theta)

180 degrees points in the negative x direction. However, sin gives us the y value, so it is unaffected by this transformation.

Sin is also an even function every 180 degrees, so it does not matter if we have a positive or negative theta. Look at a sin wave graph to see!

cos(theta) = -cos(180-theta)

180 degrees points in the negative x direction, and cos gives us the x value.

Cos is an odd function every 180 degrees.

Answer 3

Here's one example of what they want you to do:

sin(5^o) = 0.0872 allows you to find cos(10^o) by use of the following identity:

cos(2x) = 1-2sin²(x)

cos(10^o) = 1-2sin²(5^o) = 1-2•(0.0872)² = 0.9848