11D - Solving trigonometric equations
When solving trigonometric equation involving sine, cosine and tangent, there are three possible methods:
- Solving by-hand
- Solving using a CAS calculator
- Solving graphically
Solving trigonometric equations by-hand:
Use the following guide to help solve trigonometric equation:
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■ Before you start solving the expression by-hand, use formula and relationships of trigonometric functions to ensure only one trigonometric function is present. That is, only one of sine, cosine or tangent is present. |
11D - Example 1: Solving trigonometric equations
Determine all solutions for the following equation over the specified domain: \(2sin(x)-\sqrt{3}=0\) for \(x \in [0,4\pi]\).
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11D - Example 1: Video solution
11D - Example 1: Practice
Question 1: Solve the equation \(cos(x)=-\frac{\sqrt{2}}{2}\) for \(x \in [0,2\pi]\). Question 2: Solve the equation \(sin(x)=\frac{1}{2}\) for \(x \in [2\pi,4\pi]\). 11D - Example 1: Solutions
Question 1: \(x=\frac{3\pi}{4}\) and \(x=\frac{5\pi}{4}\). Question 2: \(x=\frac{13\pi}{6}\) and \(x=\frac{17\pi}{6}\). |
11D - EXAMPLE 2: General solution to trigonometric equations
Determine the general solution to the following equation:
\[cos(2x)=-\frac{1}{2}\]
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11D - Example 2: Video solution
11D - Example 2: Practice
Question 1: Find the general solution for \(sin(\frac{x}{3})=\frac{\sqrt{2}}{2}\). Question 2: Find the general solution for \(2cos(\frac{3\pi}{2})=\sqrt{3}\). 11D - Example 2: Solutions
Question 1: \(x=6\pi k+\frac{15\pi}{4}\) and \(x=6\pi k+\frac{21\pi}{4}\), \(k\in \mathbb{Z}\) Question 2: \(x=\frac{4\pi}{3} k+\frac{\pi}{9}\) and \(x=\frac{4\pi}{3} k+\frac{11\pi}{9}\), \(k\in \mathbb{Z}\) |
11D - Example 3: Solving TRIGONOMETRIC equations
Determine all solutions for the following equation over the specified domain: \(3tan(x)=\sqrt{3}\) for \(x \in [-2\pi,2\pi]\).
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11D - Example 3: Video Solutions
11D - Example 3: Practice
Question 1: Solve the equation \(tan(x)+\sqrt{3}=0\) for \(x \in [0,3\pi]\). Question 2: Solve the equation \(tan(\frac{2x}{5})=1\) for \(x \in [0,4\pi]\). 11D - Example 3: Solutions
Question 1: \(x=\frac{2\pi}{3}\) , \(x=\frac{5\pi}{3}\) and \(x=\frac{8\pi}{3}\). Question 2: \(x=\frac{5\pi}{8}\) and \(x=\frac{25\pi}{8}\). |
11D - Example 4: Solving trigonometric equations
Determine all solutions for the following equation over the specified domain:
\(2cos(x+\frac{\pi}{6})+1=0\) for \(x \in [-2\pi,\pi]\)
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11D - Example 4: Video Solution
11D - Example 4: Practice
Question 1: Solve \(sin(x-\frac{\pi}{4})=\frac{\sqrt{3}}{2} \) for \(x \in [0,2\pi] \). Question 2: Solve \(2cos(4x-\frac{\pi}{2})-\sqrt{2}=0 \) for \(x \in [0,\pi] \). 11D - Example 4: Solutions
Question 1: \(x=\frac{7\pi}{12}\) and \(x=\frac{11\pi}{12}\). Question 2: \(x=\frac{\pi}{16}\), \(x=\frac{3\pi}{16}\), \(x=\frac{9\pi}{16}\), and \(x=\frac{11\pi}{16}\). |
Solving trigonometric equations using CAS:
Please be aware of the following when solving a trigonometric equation when using a CAS calculator:
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■ Unless otherwise specified, all answers should be given in exact form. Therefore, the calculator should be in standard mode. For trigonometric equations, this will generally give the answer in terms of \(\pi \). |
Solving trigonometric equations graphically:
The CAS calculator can also be used to solve an equation graphically.
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11D - Example 5: Using the CAS calculator to solve trigonometric equations
Solve \(sin(4x)=\frac{1}{2}\), where \(x \in [0,\pi]\)
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11D - Example 5: Video SOlution
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Additional Exercises
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Topic Worksheets
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Other Resources
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Solving trigonometric equations - Worksheet A
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Solving trigonometric equations - Worksheet B
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