IB Mathematics AA Solving trigonometric equations Study Notes
IB Mathematics AA Solving trigonometric equations Study Notes
IB Mathematics AA Solving trigonometric equations Study Notes Offer a clear explanation of Solving trigonometric equations , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Solving trigonometric equations.
Solving Trigonometric Equations
Solving Trigonometric Equations
Trigonometric equations can have multiple solutions within a finite interval due to the periodic nature of sine, cosine, and tangent functions.
- Analytical methods: Solve the equation algebraically, then check for all solutions in the given interval.
- Graphical methods: Plot both sides of the equation on the same axes and identify intersection points within the specified domain.
GDC / Desmos steps (general method):
- Enter the left-hand side (LHS) as
Y1. - Enter the right-hand side (RHS) as
Y2. - Adjust the viewing window to cover the specified domain (e.g.,
0 ≤ x ≤ 2πor-π ≤ x ≤ 3π). - Use the intersection or trace feature to find solutions where
Y1 = Y2. - Record all solutions visible within the domain.
Example
Solve \( 2 \sin x = 1 \) for \( 0 \le x \le 2\pi \).
▶️Answer/Explanation
\( \sin x = \frac{1}{2} \)
Solutions: \( x = \frac{\pi}{6}, \frac{5\pi}{6} \)
Example
Solve \( 2 \sin 2x = 3 \cos x \) for \( 0^\circ \le x \le 180^\circ \).
▶️Answer/Explanation
\( \sin 2x = 2 \sin x \cos x \)
\( 4 \sin x \cos x = 3 \cos x \)
\( \cos x (4 \sin x – 3) = 0 \)
\( \cos x = 0 \Rightarrow x = 90^\circ \)
\( \sin x = \frac{3}{4} \Rightarrow x \approx 48.6^\circ, 131.4^\circ \)
Solutions: \( x = 48.6^\circ, 90^\circ, 131.4^\circ \)
Example
Solve \( 2 \tan(3(x – 4)) = 1 \) for \( -\pi \le x \le 3\pi \), give general solution.
▶️Answer/Explanation
\( \tan(3(x – 4)) = \frac{1}{2} \)
\( 3(x – 4) = \tan^{-1}\left( \frac{1}{2} \right) + n\pi \)
\( 3(x – 4) = 0.464 + n\pi \)
\( x = 4 + \frac{0.464 + n\pi}{3} \)
Substitute integers for \( n \) to find all \( x \) in range \( -\pi \le x \le 3\pi \).
Equations Leading to Quadratic Trig Equations
Equations Leading to Quadratic Trig Equations
Sometimes trigonometric equations can be rewritten as quadratic equations in sinx, cosx, or tanx. These can be solved using standard algebraic methods (e.g., factorization, quadratic formula).
Steps:
- Express the equation in quadratic form: \( a u^2 + b u + c = 0 \), where \( u \) is \( \sin x \), \( \cos x \), or \( \tan x \).
- Solve the quadratic equation for \( u \).
- Find corresponding \( x \)-values in the given domain.
- List all solutions within the domain.
Example
Solve \( 2 \sin^2 x + 5 \cos x + 1 = 0 \) for \( 0 \le x \le 4\pi \).
▶️Answer/Explanation
Use identity \( \sin^2 x = 1 – \cos^2 x \).
Equation becomes: \( 2(1 – \cos^2 x) + 5 \cos x + 1 = 0 \)
\( 2 – 2 \cos^2 x + 5 \cos x + 1 = 0 \)
\( -2 \cos^2 x + 5 \cos x + 3 = 0 \)
\( 2 \cos^2 x – 5 \cos x – 3 = 0 \)
Solve quadratic: \( \cos x = \frac{5 \pm \sqrt{25 – 4 \cdot 2 \cdot (-3)}}{4} \)
\( \cos x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm 7}{4} \)
\( \cos x = 3 \) (not valid) or \( \cos x = -\frac{1}{2} \)
Valid solutions: \( \cos x = -\frac{1}{2} \Rightarrow x = \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3} \)
Example
Solve \( 2 \sin x = \cos 2x \) for \( -\pi \le x \le \pi \).
▶️Answer/Explanation
Use identity \( \cos 2x = 1 – 2 \sin^2 x \).
\( 2 \sin x = 1 – 2 \sin^2 x \)
\( 2 \sin^2 x + 2 \sin x – 1 = 0 \)
Let \( u = \sin x \): \( 2u^2 + 2u – 1 = 0 \)
\( u = \frac{-2 \pm \sqrt{4 – 4 \cdot 2 \cdot (-1)}}{4} \)
\( u = \frac{-2 \pm \sqrt{12}}{4} = \frac{-2 \pm 2 \sqrt{3}}{4} = \frac{-1 \pm \sqrt{3}}{2} \)
Solutions: \( \sin x = \frac{-1 + \sqrt{3}}{2}, \frac{-1 – \sqrt{3}}{2} \) (only values in range \(-1\) to \(1\) valid)
Find \( x \)-values for valid solutions in \( -\pi \le x \le \pi \)