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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section TF-3 Trigonometric Equations

Worksheet Solving Trigonometric Equations

For this outcome Specifically, for this outcome, students should be able to:
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Fundamental Trigonometric Identities
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These identities are essential for solving trigonometric equations and simplifying expressions:
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  • Pythagorean Identities: \(\sin^2(x) + \cos^2(x) = 1\text{,}\) \(\tan^2(x) + 1 = \sec^2(x)\text{,}\) \(1 + \cot^2(x) = \csc^2(x)\)
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  • Reciprocal Identities: \(\csc(x) = \frac{1}{\sin(x)}\text{,}\) \(\sec(x) = \frac{1}{\cos(x)}\text{,}\) \(\cot(x) = \frac{1}{\tan(x)}\)
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  • Quotient Identities: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\text{,}\) \(\cot(x) = \frac{\cos(x)}{\sin(x)}\)
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Basic Trigonometric equations
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Basic trigonometric equations involve solving for an unknown angle in a trigonometric function. For example, solving \(\sin(x) = \frac{1}{2}\) involves finding all angles \(x\) where the sine function equals \(\frac{1}{2}\text{.}\) To do this we first find the principal value using the inverse sine function, then we consider all coterminal angles. If we are solving in a given interval, we only consider the angles within that interval.
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Sum and Difference Identities
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Sum and Difference Identities allow us to express the sine, cosine, and tangent of a sum or difference of angles in terms of the sine, cosine, and tangent of the individual angles. These identities are useful for solving equations that involve sums or differences of angles. For example, the identity for sine is \(\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)\text{.}\) By applying these identities, we can rewrite trigonometric equations in a form that is easier to solve.
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The sum and difference identities are:
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Product-to-Sum and Sum-to-Product Identities
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Product-to-Sum and Sum-to-Product Identities allow us to convert products of sine and cosine functions into sums or differences, and vice versa. These identities are useful for solving equations that involve products of trigonometric functions. For example, the product-to-sum identity for sine is \(\sin(a)\sin(b) = \frac{1}{2}[\cos(a - b) - \cos(a + b)]\text{.}\) By applying these identities, we can rewrite trigonometric equations in a form that is easier to solve.
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The product-to-sum identities are:
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Half-Angle and Double Angle Identities
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Half-Angle and Double Angle Identities allow us to express the sine, cosine, and tangent of half or double an angle in terms of the sine, cosine, and tangent of the original angle. These identities are useful for solving equations that involve half or double angles. For example, the double angle identity for sine is \(\sin(2a) = 2\sin(a)\cos(a)\text{.}\) By applying these identities, we can rewrite trigonometric equations in a form that is easier to solve.
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The half-angle and double angle identities are:
  • \(\displaystyle \sin(2a) = 2\sin(a)\cos(a)\)
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  • \(\displaystyle \cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)\)
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  • \(\displaystyle \tan(2a) = \frac{2\tan(a)}{1 - \tan^2(a)}\)
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  • \(\displaystyle \sin\left(\frac{a}{2}\right) = \pm\sqrt{\frac{1 - \cos(a)}{2}}\)
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  • \(\displaystyle \cos\left(\frac{a}{2}\right) = \pm\sqrt{\frac{1 + \cos(a)}{2}}\)
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  • \(\displaystyle \tan\left(\frac{a}{2}\right) = \pm\sqrt{\frac{1 - \cos(a)}{1 + \cos(a)}} = \frac{\sin(a)}{1 + \cos(a)} = \frac{1 - \cos(a)}{\sin(a)}\)
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Using Identities to Solve Trigonometric Equations
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Trigonometric identities can be used to simplify equations and make them easier to solve. For example, if we have an equation like \(\sin^2(x) + \cos^2(x) = 1\text{,}\) we can use the Pythagorean identity to rewrite it as \(1 = 1\text{,}\) which is always true. This means that the original equation is satisfied for all values of \(x\text{.}\) In other cases, we may need to use identities to rewrite the equation in a form that allows us to apply inverse trigonometric functions or algebraic techniques, like factoring or substitution.
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Worksheet TF-3 Trigonometric Equations

Inverse Trigonometric Functions
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Fundamental Trigonometric Identities
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Basic Trigonometric Equations
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Sum and Difference Formulas
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Double Angle Formulas
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Half Angle Formulas
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More Trigonometric Equations
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Section TF-3 Rubric

Worksheet Worksheet

Before attempting the TF-3 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 24. TF-3 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner. No algebraic errors but some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and provides a clear solution. Problems have a clear beginning, middle and end. Work progresses clearly from one step to the next. The work provided uses algebraic methods and provides a solution. Parts of the mathematical structure are missing or steps in the progress of the solution are missing. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution and it is unclear what methods are being used.
Solving Basic Trigonometric Equations Correctly isolate the trigonometric function, use the inverse to find the reference angle, then find all requested solutions, in radians, to the equation. Correctly isolate the trigonometric function, use the inverse to find the reference angle, then find some requested solutions, in radians, to the equation. Errors in isolating the trigonometric function or in finding the reference angle.  Some solutions are given either in radians or in degrees. No evidence shown of being able to solve trigonometric equations.
Solving Advanced Trigonometric Equations Correctly use an identity or property to rewrite the trigonometric equation to isolate the trigonometric function(s), use the inverse to find the reference angle, then find all requested solutions, in radians, to the equation. Correctly use an identity or property to rewrite the trigonometric equation to isolate the trigonometric function(s),, use the inverse to find the reference angle, then find some requested solutions, in radians, to the equation. Errors in rewriting the trigonometric equation, isolating the trigonometric function(s), or in finding the reference angle.  Some solutions are given either in radians or in degrees. No evidence shown of being able to rewrite or solve trigonometric equations.
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Worksheet TF-3 Trigonometric Equations

This outcome covers solving trigonometric equations. You should be able to solve equations involving sine, cosine, tangent, and their reciprocals. You should also be able to find all solutions to these equations and solutions on specific intervals. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
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1.

(a)
Find all solutions to the equation: \(\csc^2\theta-4=0\)
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Hint.
Isolate the trigonometric function then use the inverse to find the angles.
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Answer.
\(\theta=\frac{\pi}{6}+n\pi\) and \(\theta=\frac{5\pi}{6}+n\pi\text{,}\) where \(n\) is any integer.
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Solution.
Start by isolating the trigonometric function:
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\(\csc^2\theta=4\)
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Take the square root of both sides:
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\(|\csc\theta|=2\)
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This means:
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\(\csc\theta=2\) or \(\csc\theta=-2\)
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Recall that \(\csc\theta=\frac{1}{\sin\theta}\text{,}\) so:
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\(\sin\theta=\frac{1}{2}\) or \(\sin\theta=-\frac{1}{2}\)
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Find the angles where sine equals these values:
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\(\theta=\frac{\pi}{6}+n\pi\) and \(\theta=\frac{5\pi}{6}+n\pi\text{,}\) where \(n\) is any integer.
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2.

(a)
Find all solutions to the equation \(\sqrt{3}\tan3\theta+1=0\) on the interval \([0,2\pi)\)
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Hint.
Start by isolating the trigonometric function. Then use the inverse tangent to find the angles.
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Answer.
\(\theta \in \left\{\frac{5\pi}{18}, \frac{11\pi}{18}, \frac{17\pi}{18}, \frac{23\pi}{18}, \frac{29\pi}{18}, \frac{35\pi}{18}\right\}\)
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Solution.
Start by isolating the trigonometric function:
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\(\sqrt{3}\tan3\theta=-1\)
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Divide both sides by \(\sqrt{3}\text{:}\)
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\(\tan3\theta=-\frac{1}{\sqrt{3}}\)
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Take the inverse tangent of both sides:
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\(3\theta=\arctan\left(-\frac{1}{\sqrt{3}}\right)\)
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This gives us:
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\(3\theta=\frac{5\pi}{6}+n\pi\text{,}\) where \(n\) is any integer.
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Therefore:
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\(\theta=\frac{5\pi}{18}+\frac{n\pi}{3}\text{,}\) where \(n\) is any integer.
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