Solving Trigonometric Equations Using Identities: Videos & Practice Problems

To solve complex linear trigonometric equations, utilize trigonometric identities to simplify expressions into a single function. For example, the equation sec2⁢θ-1 over tan(θ) equals 1 can be simplified using the Pythagorean identity to find tan(θ) equals 1. Similarly, for sin(2θ) over cos(-θ) equals 1, apply double angle and even-odd identities to isolate sin(θ) equals 1/2, yielding solutions θ=π6+2πn and θ=5π6+2πn.

Solve Trig Equations Using Identity Substitutions

When solving linear trigonometric equations, the goal is to find an angle \( \theta \) that satisfies the equation. As equations become more complex, involving multiple trigonometric functions, the process requires the application of trigonometric identities to simplify the expressions. For instance, consider the equation:

\[\frac{\sec^2(\theta) - 1}{\tan(\theta)} = 1\]

To solve this, we can utilize the Pythagorean identity, which states that \( \sec^2(\theta) - 1 = \tan^2(\theta) \). By substituting this identity into the equation, we rewrite it as:

\[\frac{\tan^2(\theta)}{\tan(\theta)} = 1\]

Next, we can simplify the left side by canceling one \( \tan(\theta) \) from the numerator and denominator, resulting in:

\[\tan(\theta) = 1\]

From here, we can determine the angles \( \theta \) on the unit circle where the tangent function equals 1, which occurs at \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer, accounting for all possible solutions.

In another example, consider the equation:

\[\frac{\sin(2\theta)}{\cos(-\theta)} = 1\]

Here, we can apply the double angle identity for sine, which gives us:

\[\sin(2\theta) = 2\sin(\theta)\cos(\theta)\]

Thus, the equation becomes:

\[\frac{2\sin(\theta)\cos(\theta)}{\cos(\theta)} = 1\]

Using the even-odd identity for cosine, we can rewrite \( \cos(-\theta) \) as \( \cos(\theta) \). This allows us to simplify the equation to:

\[2\sin(\theta) = 1\]

To isolate \( \sin(\theta) \), we divide both sides by 2:

\[\sin(\theta) = \frac{1}{2}\]

The angles \( \theta \) that satisfy this equation on the unit circle are \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \). Since the problem asks for all solutions, we express these as:

\[\theta = \frac{\pi}{6} + 2\pi n \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2\pi n\]

By mastering the use of trigonometric identities, we can effectively solve more complex trigonometric equations and find all possible solutions.

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. $\theta=\frac{\pi}{3}+\pi n,\frac{2\pi}{3}+\pi n$

Answer: (D)

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. $\theta=\frac{3\pi}{4},\frac{7\pi}{4}$

Answer: (A)

Here’s what students ask on this topic:

How do you solve trigonometric equations using Pythagorean identities?

To solve trigonometric equations using Pythagorean identities, first identify parts of the equation that match known identities. For example, if you have ${\sec }^2\theta -\; 1$, recognize that it can be rewritten using the identity ${\sec }^2\theta =\; 1\; +{\tan }^2\theta$. This simplifies to ${\tan }^2\theta$. Substitute this back into the equation to reduce it to a single trigonometric function, which can then be solved using standard methods.

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

Double angle identities express trigonometric functions of double angles (2θ) in terms of single angles (θ). For example, $\sin \left(2\theta \right)=\; 2\sin \theta \cos \theta$. To use them in solving equations, replace the double angle term with its identity. For instance, in the equation $\frac{\sin }{\left(}=\; 1$, use the double angle identity to rewrite the numerator, then simplify and solve for θ.

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

Even-odd identities help simplify trigonometric functions with negative arguments. For example, $\cos (-\theta )=\cos \left(\theta \right)$ and $\sin (-\theta )=\; -\sin \left(\theta \right)$. To use them, replace the negative argument with its equivalent positive form. For instance, in the equation $\frac{\sin }{\left(}=\; 1$, rewrite $\cos (-\theta )$ as $\cos \left(\theta \right)$ to simplify the equation.

What are the steps to solve the equation $\frac{\sin }{\left(}=\; 1$?

To solve $\frac{\sin }{\left(}=\; 1$, follow these steps:

1. Use the double angle identity: $\sin \left(2\theta \right)=\; 2\sin \theta \cos \theta$.
2. Rewrite the equation: $\frac{2}{\sin }=\; 1$.
3. Use the even-odd identity: $\cos (-\theta )=\cos \left(\theta \right)$.
4. Simplify: $\frac{2}{\sin }=\; 1$ becomes $2\sin \theta =\; 1$.
5. Solve for $\sin \theta$: $\sin \theta =\frac{1}{2}$.
6. Find all solutions: $\theta =\frac{\pi }{6}+\; 2\pi n$ and $\theta =\frac{5}{\pi }+\; 2\pi n$.

How do you find all solutions to a trigonometric equation?

To find all solutions to a trigonometric equation, first solve for the principal solutions within one period. For example, if $\sin \theta =\frac{1}{2}$, the principal solutions are $\theta =\frac{\pi }{6}$ and $\theta =\frac{5}{\pi }$. To find all solutions, add the period of the function multiplied by an integer $n$. For sine and cosine, the period is $2\pi$, so the general solutions are $\theta =\frac{\pi }{6}+\; 2\pi n$ and $\theta =\frac{5}{\pi }+\; 2\pi n$.