Solving Trigonometric Equations Using Identities - Online Tutor, Practice Problems & Exam Prep

To solve linear trigonometric equations, we aim to find an angle θ that satisfies the equation. By applying trigonometric identities, such as the Pythagorean identity, we can simplify complex equations into a single trigonometric function. For example, transforming secant and tangent functions leads to a linear equation. Additionally, using double angle identities and even-odd identities helps in further simplification. Ultimately, we identify angles on the unit circle and express all solutions as θ = angle + 2πn, ensuring a comprehensive understanding of trigonometric relationships.

Solve Trig Equations Using Identity Substitutions

Hey, everyone. When solving linear trigonometric equations, we want to find an angle theta that makes our equation true. And now that we're tasked with solving more complicated trigonometric equations, we still want to do the same thing. We want to find an angle theta that makes our equation true. But now that these equations have more than just one trig function, it's not quite so simple because how are we going to find an angle theta for which the equation sec2θ-1 over tanθ is equal to 1? I don't know how to do that just using the unit circle. But now that we know a bunch of different trigonometric identities, we can use those identities to rewrite these equations in terms of just one trig function. Then we're just left with a linear trigonometric equation that we already know how to solve. So here, I'm going to walk you through exactly how to do that. Let's go ahead and get started.

Now here we have the equation sec2θ-1 over tanθ is equal to 1. Seeing that I have multiple trig functions here tells me that I need to use my trigonometric identities to rewrite this in terms of just one trig function. Now we're going to keep using our strategies for simplifying here. Scanning this for identities, I see in my numerator this sec2θ-1. Whenever I have a squared trig function, I should be thinking about using my Pythagorean identities here. Seeing that I have this sec2θ-1, I can use my Pythagorean identity to rewrite this, giving that this is just tan2θ over tanθ is equal to 1, having used that Pythagorean identity. From here I can cancel some stuff because that tangent in the bottom will cancel with one of my tangents in the top, leaving me with just tanθ=1. So we started with multiple trig functions and then we used identities to get down to just one trig function. And from here, we can just find our angles theta on the unit circle for which this is true and then add this factor of pi in in order to account for all possible solutions.

Now not all of these equations are going to be quite so simple, and we're not always just going to use our Pythagorean identities. So let's go ahead and walk through another example together. Here, we're asked to find all solutions to the equation sin2θ over cos-θ is equal to 1. Here I want to be constantly scanning for identities. And something that you might notice here is that we have a sine over a cosine. So your first instinct might be to say, okay, wouldn't that just be the tangent? But we can't actually do that here because these arguments are not the same. In my numerator, I have 2θ in my argument and in my denominator, I have the negative θ. So I can't use that identity. I need to think of another way to simplify this.

Well, looking at that numerator, I have the sin2θ. So having 2θ in that argument, what identity should we be using? Well, here we can rewrite the top using a double angle identity since we have that 2θ in the argument. So using that identity here, I end up with 2sinθcosθ over cos-θ, keeping that the same for now, is equal to 1, having rewritten the top there. Now looking at this, I see in my denominator I have this negative θ.

Find all solutions to the equation.

$\left(\cos\theta+\sin\theta\right)\left(\cos\theta-\sin\theta\right)=-\frac12$

• A. $\theta=\frac{5\pi}{12}+2\pi n,\frac{7\pi}{12}+2\pi n$
• B. $\theta=\frac{2\pi}{3}+2\pi n,\frac{4\pi}{3}+2\pi n$
• C. $\theta=\frac{\pi}{3}+2\pi n,\frac{2\pi}{3}+2\pi n$
• D. $\theta=\frac{\pi}{3}+\pi n,\frac{2\pi}{3}+\pi n$

Find all solutions to the equation where 0 ≤ $\theta$ ≤ $2\pi$.

$\sin\theta\cos\left(2\theta\right)-\sin\left(2\theta\right)\cos\theta=\frac{\sqrt2}{2}$

• A. $\theta=\frac{5\pi}{4},\frac{7\pi}{4}$
• B. $\theta=\frac{\pi}{4},\frac{3\pi}{4}$
• C. $\theta=\frac{5\pi}{4}+2\pi n,\frac{7\pi}{4}+2\pi n$
• D. $\theta=\frac{3\pi}{4},\frac{7\pi}{4}$

Here’s what students ask on this topic:

How do you solve trigonometric equations using Pythagorean identities?

To solve trigonometric equations using Pythagorean identities, you first identify parts of the equation that match known identities, such as ${\sin }^2\theta +{\cos }^2\theta =\; 1$. For example, if you have ${\sec }^2\theta -\; 1$, you can use the identity ${\sec }^2\theta -\; 1\; ={\tan }^2\theta$. This simplifies the equation to a single trigonometric function, making it easier to solve. Once simplified, solve for the angle $\theta$ using the unit circle and express all solutions as $\theta =\mathrm{angle}+\; 2\mathrm{\pi n}$.

What are double angle identities and how are they used in solving trigonometric equations?

Double angle identities express trigonometric functions of double angles (2$\theta$) in terms of single angles. For example, $\sin 2\theta =\; 2\sin \theta \cos \theta$ and $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$. These identities are useful for simplifying equations where the argument is a double angle. For instance, if you have $\sin 2\theta$, you can rewrite it as $2\sin \theta \cos \theta$, making it easier to solve for $\theta$. This helps in reducing the complexity of the equation and finding the solutions on the unit circle.

How do you use even-odd identities to solve trigonometric equations?

Even-odd identities help simplify trigonometric functions with negative arguments. For example, $\cos (-\theta )\; =\cos \theta$ (even function) and $\sin (-\theta )\; =\; -\sin \theta$ (odd function). If you encounter a term like $\cos (-\theta )$ in an equation, you can replace it with $\cos \theta$. This simplification makes it easier to solve the equation. For instance, in the equation $\sin 2\theta /\cos (-\theta )\; =\; 1$, you can rewrite it as $2\sin \theta \cos \theta /\cos \theta =\; 1$, which simplifies to $2\sin \theta =\; 1$.

How do you find all solutions to a trigonometric equation?

To find all solutions to a trigonometric equation, first simplify the equation using trigonometric identities to isolate a single trigonometric function. Solve for the angle $\theta$ using the unit circle. For example, if $\sin \theta =\; 1/2$, the solutions are $\theta =\pi /6$ and $\theta =\; 5\pi /6$. To find all solutions, add the period of the function (2$\pi$ for sine and cosine) multiplied by an integer $n$. Thus, the general solutions are $\theta =\pi /6\; +\; 2\mathrm{\pi n}$ and $\theta =\; 5\pi /6\; +\; 2\mathrm{\pi n}$, where $n$ is any integer.

What are some common mistakes to avoid when solving trigonometric equations using identities?

Common mistakes to avoid include: 1) Not correctly applying trigonometric identities, such as confusing Pythagorean identities. 2) Failing to account for all solutions by not adding the period of the function (e.g., $2\pi n$ for sine and cosine). 3) Incorrectly simplifying expressions, such as canceling terms improperly. 4) Ignoring the domain of the function, which can lead to extraneous solutions. Always double-check your work and ensure you use the correct identities and account for all possible solutions.