Main content

## Precalculus

### Course: Precalculus > Unit 2

Lesson 8: Sinusoidal models# Trigonometric equations review

Review your trigonometric equation skills by solving a sequence of equations in increasing complexity.

## Practice set 1: Basic equations

### Example: Solving $\mathrm{sin}(x)=0.55$

Let's use the calculator and round to the nearest hundredth.

*(We are using radians.)*

We can use the identity $\mathrm{sin}(\pi -\theta )=\mathrm{sin}(\theta )$ to find the second solution within $[0,2\pi ]$ .

We use the identity $\mathrm{sin}(\theta +2\pi )=\mathrm{sin}(\theta )$ to extend the two solutions we found to all solutions.

Here, $n$ is any integer.

### Check your understanding

*Want to try more problems like this? Check out this exercise.*

## Practice set 2: Advanced equations

### Example: Solving $16\mathrm{cos}(15x)+8=2$

First, we isolate the trig expression:

Use the calculator and round to the nearest thousandth:

Use the identity $\mathrm{cos}(\theta )=\mathrm{cos}(-\theta )$ to find that the second solution within $[-\pi ,\pi ]$ is $-1.955$ .

Use the identity $\mathrm{cos}(\theta )=\mathrm{cos}(\theta +2\pi )$ to find all the solutions to our equation from the two angles we found above. Then we solve for $x$ (remember that our argument is $15x$ ):

Similarly, the second solution is $x=-0.130+n\cdot {\displaystyle \frac{2\pi}{15}}$ .

### Check your understanding

*Want to try more problems like this? Check out this exercise.*

## Want to join the conversation?

- Maybe I've overlooked it, but there doesn't seem to be a good explanation anywhere of all the equivalences between sine and cosines of "related" angles. The explanation for practice set 1 says "We can use the identity sin(pi−theta) = sin(theta)" but that's only one and it appears to work only for positive thetas.(26 votes)
- You should watch the videos on the "unit circle" definitions of the trig functions. After you see those, there are about 10 important trig identities which become self-evident, like sin(-theta) = -sin(theta) and so on. Just think of radii intersecting a unit circle, and think of the ways those radii can be rotated and reflected and how that will affect their distance from the x-axis and y-axis.(14 votes)

- For Practice set 3: Word problems, I keep getting t = 250.4352 + 365n and t = 1550.7676 + 365n.

I followed all the steps but I keep getting the wrong answer. Can anyone please tell me what I'm doing wrong?(6 votes)- I don't know where you went wrong from your answer, but let me help you with the process of finding the answer, step by step, so that you can see where you've gone astray.

1) What is the question ?

We're asked to find the "First day after the equinox of spring", where the length of the day is = 750minutes (so L(t)=750).

Let's clarify the "First day after the start of spring" bit, the model start where t = 0 = 1st day after the equinox of spring, so it's just a fancy way of saying "the first day after t=0"

2) Simplifying the equation

750=52 sin(2pi / 365 * t) +728

-728 on both sides

22=52 sin(2pi / 365 * t)

divide both side by 52

22/52 = sin(2pi / 365 * t)

Simplifying 22/52 = 2x11/2x26 = 11/26

11/26 = sin(2pi / 365 * t)

2) using arcsin, a mini recap: if sin(x) = y, then arcsin(y)=x. x is an angle and y is the ratio between opposite side and the hypotenuse (y=opposite/hypotenuse) and there are usually 2 points where sin(x)=y, unless it's 0, 90, 180, 270 or 360 then there's one point only (0, 1 or -1)

11/26 = sin(2pi / 365 * t)

arcsin(11/26)= 2pi / 365 * t

arcsin(11/26)=0.44rad (=25°)

Here our first answer is in the first quadrant, and sin is the y axis on the unit circle, so if we trace a line between the point at 25° on the circle and the y axis, we'll get the second answer on the circle (the second angle whose sin = 11/26)

To get it we do pi - 0.44rad = 2.70rad, (180°-25°=155°)

2pi / 365 * t = 0.44rad + 2pi * n

and 2pi / 365 * t = 2.70rad + 2pi * n

3) Solving for t by multiplying both sides by 365/2pi (a/b * b/a = ab / ab = 1)

t= 0.44 * 365/2pi + 2pi * 365/2pi * n = 25.37 + 365n

and

t= 2.70 * 365/2pi + 2pi * 365/2pi * n = 156.85 + 365n

4) Figuring which answer we need

we got t = 25.37 + 365n and t = 156.85 + 365n

we want the first answer (first day of after the equinox of spring, which we said earlier was t=0)

So the answer is t = 25.37 + 365n, we don't really care about next year so t = 25.37, and the answer is asking us to round it to the nearest whole day so t=25!

Done, the answer is 25!

Bonus =

It really helps if you draw a unit circle or the sinusoidal model to visualize what's going on, and where your answers are.

On the unit circle t=0 (spring) would be at 0°, summer at 90° etc, also the "sin" is the vertical axis and "cos" the horizontal axis, so you know where your second answers are if you get confused.

Now to draw a sinusoidal model, let's use our example here:

L(t)=52sin(365/2π*t)+728

To draw a sinusoidal model (fancy word for wavy function), just draw a x and a y axis (just the first quadrant, no need for -x or -y here)

Grab a random point at the middle of your "y" axis, that's "728"

now pick a point above 728 (728 will be the middle of the wave, the point above will be it's maximum value) that's 728+52 =780

Now pick a point below 728, the same distance between the point "780" you made earlier, and that will be 728-52=676

Why +52 and -52 ? -1<=sin(x)<=1 so sin(whatever) * 52 will be between -52 and +52

Ok, we're ready to draw our wave, now sin(0) = 0 and its in the middle of -1 and 1, and sin(1°) is greater than 0 so the wave starts by going up.

Here I made a graph of what it looks like, https://www.desmos.com/calculator/0ax1151tpa

because long explanation just make things more confusing, and pictures > words.

Hope that helped ! Sorry for the long answer, no potatoes sadly(33 votes)

- hi, i don't understand why in the second practice question we use sintheta= sin(-180-theta) and not +180-theta. i thought the identity was sintheta=sin(pi-theta) so why would we use -180-theta in degrees? thanks!(7 votes)
- Hello!

This confused me for a while too, but I can give you a brief overview of why you flip the sign. If you look at the problems, you'll notice that (180 -theta) only becomes (-180 - theta) when theta is negative. (The value I am talking about is the one you get after you apply arcsin or arccos to both sides) This is because when you are going around the unit circle, negatives tend to screw up the direction you want to go (you're going clockwise instead of counterclockwise with a negative) to combat this, the sign is flipped to become -180-theta (though because theta is negative it then becomes -180 + theta. Double negatives turn into a positive)

Hope this helps!(6 votes)

- I have been having problems with the math, specifically the word problems involving inverse trig. Every time I have tried to get the answer, my calculator ends up getting it wrong despite my entering it in correctly.

For example, on the question where I have to solve for t:

L(t) models the length of each day (in minutes) in Manila, Philippines(t) days after the spring equinox. Here, (t) is entered in radians. L(t)=52sin((2pi/365)t)+728, find the first day after the spring equinox that the day length is 750 minutes

I end up with t= (365/2pi)(2.7048+n*2pi)= 250.435, but every time I check my incorrect answer, it gives me the same exact equation I got but with 25 as the answer (rounded to the nearest whole number) instead-- an entire decimal place lower. I even follow the steps using my calculator when I'm checking and still get 250.

I don't know what exactly I'm doing wrong and I would appreciate any help I could get, thanks!(5 votes)- Can you write out each step perhaps? or if I do it:

52sin((2pi/365)t)+728 = 750 First subtract 728 from both sides

52sin((2pi/365)t) = 22 Divide both sides by 52

sin((2pi/365)t) = 22/52 = .42307

sine of an angle is the y value of the radius when it is at that angle, so it is even less than sin(pi/6), so we know that at least. This also means it is in the domain of arcsin, which is good.

sin((2pi/365)t) = 22/52 = .42307 inverse sine or arcsin of both sides

2pi/365 t = arcsin(22/52) divide both sides by 2pi/365

t = arcsin(22/52)365/(2pi)

t = 25.3766

I did notice I got 250.4572 when I plugged in the divided by 2pi into my calculator. If this is where it happens to you, put the 2pi in parenthesis, or do them seperately. It's kind of a good rule to put as much as you can in parenthesis on a calculator.

Let me know if this did not help.(6 votes)

- I'm confused about using trig identities. For example, in some of the practice problems the identity cosx = cos-x is used. In others the identity cosx = cos(2pi - x) is used. It seems like I end up in the same location on the unit circle either way. But I get the "bleep" for "wrong answer" if I do not choose the identity that the person presenting the solution used. Is there some rule to apply here?(8 votes)
- They are the same.

cos(2pi - x) = cos(-x)

You have to adjust the`n`

factor to find a positive value.

Let's say you find:**arccos(x) = 1.5 + 2n**

and**arccos(x) = -1.5 + 2n**

The**first positive**value isn't 1.5. It's**0.5**, because of (-1.5 + 2n), with n=1.(1 vote)

- In the sin demonstration above, can anyone tell me why K.A. finds the second solution over the interval [0, 2pi] , while in the cos demonstration they find the second solution over the interval [-pi, pi]?

Are these established conventions, K. A.'s convention, or just arbitrary examples?

I'd rather not get into practice or mastery problems, only to find that my answers are scored wrong simply because I did not use some unstated convention or expectation...(6 votes)- There is no simpler solution than finding the second angle that results in the same cosine by using the even function property (visualize and see why it's true). On the unit circle, the angle 1.955 is in the second quadrant. The second angle in [0, 2𝜋] with the same cosine must then be in the third quadrant. The
*easiest*way to find this second angle is to find a*coterminal*angle that is not within [0, 2𝜋]. We do this by finding the angle in [-𝜋, 𝜋] that has the same cosine by using the even function property. Then, we can translate this to the desired second solution by simply adding 2𝜋, which brings us to our coterminal second solution in [0, 2𝜋].

These are not "conventions". These are problem-solving strategies that minimize the amount of work required. Comment if you have questions!(2 votes)

- Can anyone tell me which identities I need to/should memorize? Thanks!(5 votes)
- If a kid is still taking Trig, pointing to the Putnam is premature!

Prep+participate in the AMCs first - https://www.maa.org/math-competitions/amc-1012

Low - AMC-8 Problems & Solutions on AoPS - https://goo.gl/Y6ot1w

Mid - AMC-10 Problems & Solutions on AoPS - https://goo.gl/VyJBCT

High - AMC-12 Problems & Solutions on AoPS - https://goo.gl/kDYA2M

P.S. If you're trying to learn all trig identities, this PDF is a nice one:

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

P.P.S. It's better to learn why they all work!(3 votes)

- how do i know which identity to use when solving(6 votes)
- Hi! Can anyone help me solve this:
**sin(5x)-√3cos(5x)=√3**(5 votes) - I don't know why, but for the last question when I typed 0.437*365/2pi into my calculator, it evaluated to 250 instead of 25. The decimal points are fine, and my calculator is in radians, so does anyone know what the problem is?(3 votes)
- With what you typed, the calculator followed order of operations rules. It multiplied: 0.437*365. Then, divided by 2, and last multiplied by Pi. Multiplication and division is done from left to right.

If 2Pi is meant to be the divisor, you need to use parentheses to get the multiplication of 2Pi done before the division. Enter: 0.437*365/(2pi)

Hope this helps.(4 votes)