Linear Trigonometric Equations (DP IB Analysis & Approaches (AA): HL): Revision Note

Trigonometric equations: sinx = k

How do I solve trigonometric equations?

• Rearrange the equation to isolate the trig term

  • e.g. rearrange $5\sin x=3$ to $\sin x=\frac{3}{5}$

• You can then find all the solutions using the trig graphs or the unit circle

• The steps are summarised below

sinx = k & cosx = k

• STEP 1
  Find the principal value using the inverse trig functions

  • You might have to use your GDC or your knowledge of exact trig values

    • e.g. for $\sin x=0.5$ you find ${\sin }^{-1}\left(0.5\right)=30°$

    • e.g. for $\cos x=-0.5$ you find ${\cos }^{-1}\left(-0.5\right)=120°$

• STEP 2
  Find a second angle

  • For $\sin x=k$: subtract the principal angle from 180°

    • e.g. $180°-30°=150°$

  • For $\cos x=k$: subtract the principal angle from 360°

    • e.g. $360°-120°=240°$

• STEP 3
  Find all the angles in the given interval by adding or subtracting multiples of 360° to your two angles

  • e.g. $30°+360°=390°$ and $150°+360°=510°$

  • e.g. $120°-360°=-240°$ and $240°-360°=-120°$

tanx = k

• STEP 1
  Find the principal value using the inverse trig functions

  • You might have to use your GDC or your knowledge of exact trig values

    • e.g. for $\tan x=1$ you find ${\tan }^{-1}\left(1\right)=45°$

• STEP 2
  Find all the angles in the given interval by adding or subtracting multiples of 180° to your principal angle

  • e.g. $45°+180°=225°$ and $45°+360°=405°$

Trigonometric equations: sin(ax + b) = k

How can I solve equations with transformations of trig functions?

• STEP 1
  Make the substitution $y=ax+b$

  • e.g. for $\sin \left(2x+60°\right)=\frac{\sqrt{3}}{2}$ :

    • let $y=2x+60°$

    • the equation becomes $\sin y=\frac{\sqrt{3}}{2}$

• STEP 2
  Write the interval in terms of the substitution

  • e.g. for the interval $0°\le x\le 360°$:

    • multiply each part by 2 and add 60°

    • the interval becomes $0°\le 2x+60°\le 780°$

    • this is the same as $0°\le y\le 780°$

• STEP 3
  Find all the solutions for $y$ that are in the interval

  • e.g. ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=60°$

    • $180°-60°=120°$

    • $60°+360°=420°$

    • $120°+360°=480°$

    • $60°+720°=780°$

    • $120°+780°=900°$ ⨉ is outside the interval

• STEP 4
  Find the value of $x$ for each value of $y$ using the substitution

  • $y=2x+60°\Rightarrow x=\frac{y-60°}{2}$

    • $x=0°,30°,180°,210°,360°$