Trigonometric Equations — NDA Maths

Formulas (4)

• The General-Solution Formulas · General solutions
  $$\sin\theta=\sin\alpha:\ \theta=n\pi+(-1)^n\alpha; \quad \cos\theta=\cos\alpha:\ \theta=2n\pi\pm\alpha; \quad \tan\theta=\tan\alpha:\ \theta=n\pi+\alpha; \quad \sin\theta=0:\ \theta=n\pi; \quad \cos\theta=0:\ \theta=(2n+1)\tfrac{\pi}{2}; \quad \sin^2\theta=\sin^2\alpha:\ \theta=n\pi\pm\alpha$$
• Reducing an Equation to Standard Form · Co-function reduction
  $$\cos\theta = \sin\!\left(\tfrac{\pi}{2} - \theta\right), \quad \sin\theta = \cos\!\left(\tfrac{\pi}{2} - \theta\right)$$
• Counting Solutions in an Interval · Count = solutions of the reduced equation in range
  $$\cot 2x\,\cot 3x = 1 \ \Rightarrow\ \cos 5x = 0 \ \Rightarrow\ 5x = (2n+1)\tfrac{\pi}{2}$$
• Range & Existence Conditions · Existence bound
  $$a\sin x = b \ \text{ solvable} \iff |b| \le |a|$$

Watch out for (10)

• sin and cos use DIFFERENT general forms
  → The General-Solution Formulas
• The $(-1)^n$ belongs to SINE only
  → The General-Solution Formulas
• The $\pm$ is the whole point of the cosine form
  → The General-Solution Formulas
• tan and sin share $n\pi$ but differ in the add-on
  → The General-Solution Formulas
• Squaring can add false roots
  → Reducing an Equation to Standard Form
• Dividing by $\cos x$ (or $\sin x$) can LOSE roots
  → Reducing an Equation to Standard Form
• $\sin x = 2$ has NO solution
  → Reducing an Equation to Standard Form
• Scale the interval when the angle is multiplied
  → Counting Solutions in an Interval
• Discard solutions where the equation is undefined
  → Counting Solutions in an Interval
• Count integers INCLUSIVELY across the range
  → Range & Existence Conditions

Formulas (3)

• Trig Values as Roots of a Quadratic (Vieta) · Vieta + tan-sum
  $$\tan(\alpha+\beta) = \dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} = \dfrac{-b/a}{1-c/a}$$
• Product & Sum-to-Product Forms · The product identity
  $$(1+\tan A)(1+\tan B) = 2 \iff A + B = \tfrac{\pi}{4}$$
• Logarithmic & Special Trig Equations · Reciprocal-log trick
  $$t + \tfrac{1}{t} = 2 \iff t = 1$$

Watch out for (2)

• Vieta's product is $c/a$, not $-c/a$
  → Trig Values as Roots of a Quadratic (Vieta)
• A log base must be positive and $\ne 1$
  → Logarithmic & Special Trig Equations

Formulas (2)

• Solving Two Equations Together · Common solution = intersection
  $$\{\theta : \text{eqn 1}\} \cap \{\theta : \text{eqn 2}\}$$
• Reducing a Combined System · The s-substitution
  $$s = \sin x + \cos x \ \Rightarrow\ \sin x\cos x = \tfrac{s^2-1}{2},\quad \sin 2x = s^2 - 1$$