Trigonometric equations involve trigonometric functions of unknown angles. Since trigonometric functions are periodic, an equation can have an infinite number of solutions if no specific domain or range is defined. In this topic, students learn the systematic four-step procedure to find all solutions within a given interval (in degrees or radians). Furthermore, trigonometric functions and identities are applied to solve practical real-world problems involving heights, distances, speeds, angles of elevation, and geometric configurations.
Guide students systematically through identifying the reference angle \(\alpha\) and calculating the actual angle \(\theta\) across all four quadrants.
Simplify the equation by using suitable identities if needed.
Determine the reference angle \(\alpha\) using the positive value of the trigonometric ratio.
Find the angles in the quadrants that correspond to the signs of the ratio and the adjusted range.
Write the solutions obtained within the given domain.
A systematic four-step procedure for solving trigonometric equations and applying them to real-world problems.
First, simplify it into a single trigonometric ratio of a single angle or factorise it into linear factors. Common trigonometric identities such as \(\sin 2x = 2\sin x\cos x\) or \(\cos 2x = 1 - 2\sin^2 x = 2\cos^2 x - 1\) are often used.
Next, calculate the reference angle \(\alpha\) using the absolute value of the number on a calculator (ignoring any negative sign). The calculator mode must match the requested unit (degrees or radians).
Then, identify which quadrants the angle lies in using the positive/negative sign of the original ratio (using the CAST rule). If the argument of the function is multiplied (e.g. \(2\theta\)), adjust the interval accordingly (e.g. \(0^\circ \leqslant 2\theta \leqslant 720^\circ\)) and complete multiple revolutions before dividing by the coefficient.
Finally, write down all valid angles within the original domain.
Reference angle from positive ratio
Always use the positive magnitude of the trigonometric ratio.
Compound angle identity for tangent
Used to connect combined angles in real-life geometric modelling.
Answer in your mind, then press "Check Answer".
Find all values of \(x\) for \(0^\circ \leqslant x \leqslant 360^\circ\) that satisfy the equation \(\sin 2x = -\dfrac{\sqrt{3}}{2}\).
Solve the equation \(4\sin^2 x - 3 = 0\) for \(0^\circ \leqslant x \leqslant 360^\circ\).
Using the identity \(\sec^2 A - \tan^2 A = 1\), find the exact positive value of \(\tan A\) if \(\sec^2 A + \tan^2 A = 2\).
A negative trigonometric ratio means the angle itself is negative.
In standard position (\(0^\circ \leqslant \theta \leqslant 360^\circ\)), the angle \(\theta\) is positive. The negative sign simply indicates that the terminal arm lies in a quadrant where that specific ratio has negative coordinates (e.g. Quadrants III and IV for sine).
Dividing an angle by 2 at the beginning is the same as dividing the solutions by 2.
For \(\cos 2\theta = 0.5\), students cannot find \(\cos \theta = 0.25\). One must find all values of the compound argument \(2\theta\) over the extended range \(0^\circ \leqslant 2\theta \leqslant 720^\circ\) first, and only divide the resulting angles by 2 in the final step.
Reference angle \(\alpha\) can be measured from the \(y\)-axis.
The reference angle \(\alpha\) is strictly the acute angle made between the terminal arm and the horizontal \(x\)-axis.
Objective: Identify reference angles, adjusted ranges, quadrant relationships and key identities in trigonometric applications.
1 What is the reference angle, \(\alpha\), for the equation \(\cos \theta = -0.5\) where \(0^\circ \leqslant \theta \leqslant 360^\circ\)?
2 When solving \(\sin 2x = 0.5\) for \(0^\circ \leqslant x \leqslant 360^\circ\), what is the appropriate adjusted range for the angle \(2x\)?
3 When solving a quadratic equation in terms of \(\sin x\), any solution where \(\sin x = -4\) must be ignored because the value of \(\sin x\) must lie within the range -1 to _______.
4 In Quadrant III, the relationship between the angle \(\theta\) and its acute reference angle \(\alpha\) is given by \(\theta = 180^\circ +\) _______.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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