11E - More techniques for solving:
Solving advanced trigonometric equations:
When solving more complex trigonometric questions you may need to
When solving more complex trigonometric questions you may need to
- Identify a "hidden quadratic" within the overall equation.
- Use knowledge of circular functions to ensure only one trigonometric functions is present in the equation.
11E - Example 1: Solving advanced trigonometric equations
Determine all solutions for the following equation over the specified domain:
\(2 cos^2(x)-\sqrt{3}cos(x)=0\) for \(x \in [0,2\pi]\)
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11E - Example 1: Video solution
11E - Example 1: Practice
Question 1: Solve \(2sin^2(x)=-sin(x)\) for \(x \in [0,2\pi]\). Question 2: Solve \(tan^2(x)+\sqrt{3}tan(x)=0\) for \(x \in [-2\pi,0]\). 11E - Example 1: Solutions
Question 1: \(x=\pi\), \(x=\frac{7\pi}{6}\) and \(x=\frac{11\pi}{6}\). Question 2: \(x=-\pi\), \(x=\frac{-\pi}{3}\) and \(x=\frac{-4\pi}{3}\). |
11E - Example 2: Solving advanced trigonometric equations
Determine the general solution to the following equation:
\(sin(\frac{x}{3})=cos(\frac{x}{3})\)
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11E - Example 2: Video solution
11E - Example 2: Practice
Question 1: Solve \(cos(x+\frac{\pi}{6})=sin(x+\frac{\pi}{6})\) for \(x \in [0,3\pi]\). 11E - Example 2: Solutions
Question 1: \(x=\frac{\pi}{12}\), \(x=\frac{13\pi}{12}\) and \(x=\frac{25\pi}{12}\). |
Complementary angles:
- Complementary angles sum to 90° or \(\frac{\pi}{2}\).
- Sine and cosine are complementary.
11E - Example 3: Complementary relationships
If \( sin(\theta)=0.36\), for \(0\leq \theta \leq \frac{\pi}{2}\), determine the value of
(a) \(cos(\frac{\pi}{2}-\theta)\) (b) \(cos(\frac{\pi}{2}+\theta)\) (c) \(sin(-\theta)\) |
11E - Example 3: Video solution
11E - Example 3: Practice
Question 1: If \( cos(\theta)=-0.74\), for \(\frac{\pi}{2} \leq \theta \leq \pi\), determine the value of (a) \(sin(\frac{\pi}{2}-\theta)\) (b) \(sin(\frac{\pi}{2}+\theta)\) (c) \(cos(-\theta)\) 11E - Example 3: Solutions
Question 1: (a) \(sin(\frac{\pi}{2}-\theta)=-0.74\) (b) \(sin(\frac{\pi}{2}+\theta)=-0.74\) (c) \(cos(-\theta)=-0.74\) |
11E - Example 4: Solving equations involving complementary relationships
Find \(\theta\) if \( -sin(\theta)=cos(\frac{7\pi}{8})\), where \(0\leq \theta \leq \frac{\pi}{2}\).
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11E - Example 4: Video solution
11E - Example 4: Practice
Question 1: ABC Question 2: ABC 11E - Example 4: Solutions
Question 1: ABC Question 2: ABC |
Reciprocal identities:
You should be aware of the following reciprocal identities for trigonometric functions:
You should be aware of the following reciprocal identities for trigonometric functions:
\[sec(\theta)=\frac{1}{cos(\theta)}\]
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\[cosec(\theta)=\frac{1}{sin(\theta)}\]
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\[cot(\theta)=\frac{1}{tan(\theta)}=\frac{cos(\theta)}{sin(\theta)}\]
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The third letter letter indicates which reciprocal function it is associated with. For example, seC is the reciprocal of Cosine.
11E - Example 5: Reciprocal identities
Find the exact value of \(cosec(\frac{3\pi}{4})\).
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11E - Example 5: Video solution
11E - Example 5: Practice
Question 1: ABC Question 2: ABC 11E - Example 5: Solutions
Question 1: ABC Question 2: ABC |
Using right-angled triangles to solve:
In some complex cases, it is often useful to represent the known information using a right-angled triangle. It is then possible to apply knowledge of Pythagoras’ theorem to solve for an unknown side length.
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■ Recall Pythagoras' theorem:
\(h^2=a^2+b^2\) |
11E - Example 6: Using right-angled triangles to solve
If \(sin(x)=\frac{3}{2}\) and \(\frac{\pi}{2}\leq x \leq \pi\), find the value of:
(a) \(cos(x)\) (b) \(tan(x)\) |
11E - Example 6: Video solution
11E - Example 6: Practice
Question 1: ABC Question 2: ABC 11E - Example 6: Solutions
Question 1: ABC Question 2: ABC |
The construction of the unit circle also gives rises to the Pythagorean identity.
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■ The Pythagorean identity:
\(sin^2(\theta)+cos^2(\theta)=1\) |
11E - Example 7: Using the pythagorean identity to solve
Solve the following equation for \(x\), finding all solutions for \(x \in [0,2\pi]\).
\(cos^2(x)+cos(x)=sin^2(x)\)
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11E - Example 7: Video solution
11E - Example 7: Practice
Question 1: ABC Question 2: ABC 11E - Example 7: Solutions
Question 1: ABC Question 2: ABC |
11E - Example 8: Using the pythagorean identity to solve
If \(sin(x)=a\) for \(x \in [0,\frac{\pi}{2}]\), find \(tan(x)\) in terms of \(a\).
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11E - Example 8: Video solution
11E - Example 8: Practice
Question 1: ABC Question 2: ABC 11E - Example 8: Solutions
Question 1: ABC Question 2: ABC |