You should be able to use the unit circle to find other angles with the same trig value as a given or calculated one. You can find more information on this topic in section 5.3 where the book discusses Reference Angles. The figures depict the most common situations that come up in problems of this type, however there are other things that can happen so you would be wise to practice the methods of thinking used to come up with the figures rather than memorizing these pictures. If you know how to come up with the pictures yourself, you will be able to handle a lot more situations and you will find that the material is easier for you to come back to after a break away from it, such as a summer vacation before your next math class. Learning this way will even help you when you are preparing for the final at the end of this semester because we will have a break away from trig to cover some algebra topics before the final exam.

	Sine and Cosecant First let's look at the functions sine and cosecant. As you know by now, the sine of an angle, θ is the y-value on the terminal side of the angle when measured on the unit circle counter clockwise from the positive x-axis. The cosecant of the angle θ is 1/y of the same point. Notice that the unit circles have an extra horizontal line drawn in them. This is because everywhere that the circle touches the horizontal line, the y-values are the same. Thus the values for sine and cosecant are the same at angles whose terminal sides touch the same horizontal line. Suppose you want to find the angle ρ where ρ [270°, 360°] and sin ρ = -0.7. Your calculator will give you -44.43° which is in the interval: [-90°, 0°]. Look at the figure at the left to figure out what you need to do in order to get into the desired interval. You just need to add 360° to get 315.57° which is in the desired interval. Please do not memorize these figures, but instead use the idea of looking for angles with the same y-values on the unit circle, but in the interval desired to figure out what needs to be done in any given situation. You should practice drawing your own figures as you work problems, so you can think on the circle and use it to figure out what to do.   Click here for worked out solutions to the following problems: 1. Find the angle α, where α [π/2, π] and csc α = 5. 2. Find an angle θ, where θ is in quadrant III and csc θ = -3. 3. Find the angle φ, where φ [0, π/2] and sin φ = 0.25.
	Cosine and Secant Now let's look at the functions cosine and secant. As you know by now, the cosine of an angle, θ is the x-value on the terminal side of the angle when measured on the unit circle counter clockwise from the positive x-axis. The secant of the angle θ is 1/x of the same point. Notice that the unit circles have an extra vertical line drawn in them. This is because everywhere that the circle touches the vertical line, the x-values are the same. Thus the values for cosine and secant are the same at angles whose terminal sides touch the same vertical line. Suppose you want to find the angle ρ where ρ [3π/2, 2π] and cos ρ = 0.7. Your calculator will give you 0.7954R which is in the interval: [0, π/2] so you just need to subract from 2π (look at the greenfigure in the first circle) to get 5.4878 which is in the desired interval. Use the idea of looking for angles with the same x-values on the unit circle, but in the interval desired to figure out what needs to be done in any given situation. You should practice drawing your own figures as you work problems, so you can think on the circle and use it to figure out what to do.   Click here for worked out solutions to the following problems: 4. Find the angle, γ where γ [180°, 270°] and sec γ = -10.0 5. Find an angle ε whose terminal side is in quadrant IV, where cos ε = 0.2 6. Given that cos β = 0.8764, find cos (360° - β), cos (180° - β), and cos (180° + β) without using the trig functions on your calculator and without finding β.
	Tangent and Cotangent Finally let's look at the functions tangent and cotangent. As you know by now, the tangent of an angle, θ is the y-value divided by the x-value on the terminal side of the angle when measured on the unit circle counter clockwise from the positive x-axis. The cotangent of the angle θ is x/y of the same point. Notice that the unit circles have the coterminal side extended to make a diameter of the circle. This is because everywhere that the circle touches the diameter, the ratio of the y to x-values remains the same. Thus the values for tangent and cotangent are the same at angles whose terminal sides touch the same diameter. Suppose you want to calculate cot α where cot (180° - α) = 2.5 If you draw an arbitrary angle α on a unit circle and then draw 180° - α, you will see that the ratio of the x to y value is the same except the x value has changed sign, so the answer must be -2.5. Notice we didn't need a calculator to figure this out and we didn't even need to know what the value of α is.   Click here for worked out solutions to the following problems: 7. Find δ where δ [π/2, π] and cot δ = -4. 8. Find μ where μ [180°, 270°] and tan μ = 7. 9. Given that tan ν = 0.8930, find tan (180° + ν), tan (180° - ν), and cot (90° - ν) without using the trig functions on your calculator and without finding ν.

Numbers 79 and 81 of section 5.3 are also good problems where thinking on the unit circle is helpful.

Answers:

  Back to problems 1 through 3.
1.   csc α = 5 means that sin α = 1/5. When you plug sin^-1(1/5) into your calculator, you get 0.2014^R which is in the first quadrant. We want an answer in the second quadrant, so we need: α = π - 0.2014 = 2.9402^R.

2.   csc θ = -3 means that sin θ = -1/3, so we get sin^-1(-1/3) = -0.3398^R in our calculator, but this is in quadrant IV. The θ we want is π + 0.3398 = 3.481^R = 199.471°

3.   For this one all we need to do is plug in φ = sin^-10.25 = 0.2527^R.




  Back to problems 4 through 6.
4.   sec γ = -10.0 implies cos γ = -0.1. Plugging in cos^-1(-0.1) gives 95.739°, so γ = 360° - 95.739 = 264.261°. Note: I don't really plug 95.739 back in, I just use 2^nd ANS, so that I use the answer that was already in my calculator from the prior step.

5.   ε = 360° - (cos^-10.2)° = 281.54°. You could also give -78.46° as an answer.

6.   cos (360° - β) = 0.8764, cos (180° - β) = -0.8764, and cos (180° + β) = -0.8764




  Back to problems 7 through 9.
7.   tan δ = -0.25. δ = π + tan^-1(-0.25) = 2.8966^R

8.   μ = tan^-17 + 180° = 261.8699°

9.   tan (180° + ν) = 0.8930, tan (180° - ν) = -0.8930, and cot (90° - ν) = 0.8930