Trigonometric Functions — MHT-CET Maths

Formulas (5)

• General and Principal Solutions of sin θ = k, cos θ = k, tan θ = k · The three general-solution patterns
  $$\sin\theta=\sin\alpha \Rightarrow \theta=n\pi+(-1)^n\alpha \qquad \cos\theta=\cos\alpha \Rightarrow \theta=2n\pi\pm\alpha \qquad \tan\theta=\tan\alpha \Rightarrow \theta=n\pi+\alpha$$
• Equations That Reduce to a Quadratic in One Ratio — and the Roots to Reject · Reduce, then check
  $$\cos^2 x = 1-\sin^2 x \qquad -1 \le \sin x,\ \cos x \le 1 \qquad \tan x,\ \sec x \text{ need } \cos x \ne 0$$
• a cos x + b sin x = c — the R Form, When a Solution Exists, and Roots as a Pair · The R form and the existence condition
  $$a\cos x+b\sin x=\sqrt{a^2+b^2}\,\cos(x-\phi) \qquad \text{solvable} \iff |c|\le\sqrt{a^2+b^2}$$
• Solving by Factorisation — Sum-to-Product and Multiple Angles · Sum-to-product
  $$\sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2} \qquad \cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$$
• Range Arguments — No Solution, Exponential Forms and Trigonometric Inequalities · Bounds that decide an equation
  $$\sin^4 x+\cos^4 x = 1-\tfrac12\sin^2 2x \in \left[\tfrac12,\,1\right] \qquad a^{\sin^2 x}\cdot a^{\cos^2 x}=a$$

Watch out for (6)

• Using the sine pattern for cosine
  → General and Principal Solutions of sin θ = k, cos θ = k, tan θ = k
• Counting the root the equation cannot hold
  → Equations That Reduce to a Quadratic in One Ratio — and the Roots to Reject
• Counting the integers at the boundary
  → a cos x + b sin x = c — the R Form, When a Solution Exists, and Roots as a Pair
• Dividing by a factor that can be zero
  → Solving by Factorisation — Sum-to-Product and Multiple Angles
• Counting two roots per value of sin²x instead of four
  → Range Arguments — No Solution, Exponential Forms and Trigonometric Inequalities
• A second root of an equation the key counts once
  → Range Arguments — No Solution, Exponential Forms and Trigonometric Inequalities

Formulas (4)

• The Sine Rule and the Circumradius · Sine rule
  $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$
• The Cosine Rule — an Angle from Three Sides, a Side from Two and the Included Angle · Cosine rule
  $$c^2=a^2+b^2-2ab\cos C \qquad \cos C=\frac{a^2+b^2-c^2}{2ab}$$
• Reading an Angle off a Relation Among the Sides · Target form
  $$a^2+b^2-c^2=k\,ab \iff \cos C=\frac{k}{2}$$
• The Projection Rule · Projection rule
  $$a=b\cos C+c\cos B \qquad b=c\cos A+a\cos C \qquad c=a\cos B+b\cos A$$

Watch out for (3)

• Putting the sides in the ratio of the angles
  → The Sine Rule and the Circumradius
• Computing the angle opposite the wrong side
  → The Cosine Rule — an Angle from Three Sides, a Side from Two and the Included Angle
• Losing the sign of k
  → Reading an Angle off a Relation Among the Sides

Formulas (4)

• Half-Angle Formulas in Terms of s · Half-angle formulas
  $$\tan\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \qquad \tan\frac{A}{2}\tan\frac{C}{2}=\frac{s-b}{s} \qquad \cot\frac{B}{2}\cot\frac{C}{2}=\frac{s}{s-a}$$
• Napier's Analogy — the Difference of Two Angles · Napier's analogy
  $$\tan\frac{B-C}{2}=\frac{b-c}{b+c}\cot\frac{A}{2}$$
• Right Triangle: tan of the Two Half-Angles as Roots of a Quadratic · The relation
  $$\tan\left(\tfrac{A}{2}+\tfrac{B}{2}\right)=1 \;\Rightarrow\; p+q=r \quad\text{for } px^2+qx+r=0$$
• Area — Heron's Formula and Its Consequences · Heron's formula
  $$\Delta=\sqrt{s(s-a)(s-b)(s-c)}=\tfrac12\,bc\sin A$$

Watch out for (1)

• Using the side you were given instead of the one left over
  → Half-Angle Formulas in Terms of s

Formulas (1)

• A Trigonometric Ratio of an Inverse Value — the Right-Triangle Conversion · Two conversions worth memorising
  $$\sin(\cot^{-1}x)=\cos(\tan^{-1}x)=\frac{1}{\sqrt{1+x^2}} \qquad \sin(2\sin^{-1}x)=2x\sqrt{1-x^2}$$

Reference tables (1)

Watch out for (3)

• Taking the negative out of cos⁻¹
  → Principal Ranges, Negative Arguments and f⁻¹(f(x))
• Several options can be true
  → Principal Ranges, Negative Arguments and f⁻¹(f(x))
• Doubling the ratio instead of the angle
  → A Trigonometric Ratio of an Inverse Value — the Right-Triangle Conversion

Formulas (4)

• Complementary Pairs — sin⁻¹x + cos⁻¹x = π/2 · The three complementary pairs
  $$\sin^{-1}x+\cos^{-1}x=\tan^{-1}x+\cot^{-1}x=\sec^{-1}x+\csc^{-1}x=\frac{\pi}{2}$$
• The Addition Formula — tan⁻¹x ± tan⁻¹y, 2 tan⁻¹x and Three-Term Sums · Addition formula
  $$\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}\ (xy<1) \qquad 2\tan^{-1}x=\tan^{-1}\frac{2x}{1-x^2}$$
• Simplifying by Substitution — x = tan θ, cos θ or cos 2θ · The tan θ substitution
  $$x=\tan\theta:\quad \frac{2x}{1+x^2}=\sin2\theta,\quad \frac{1-x^2}{1+x^2}=\cos2\theta,\quad \frac{2x}{1-x^2}=\tan2\theta$$
• Telescoping Sums of Inverse Tangents · The split
  $$\tan^{-1}\frac{A-B}{1+AB}=\tan^{-1}A-\tan^{-1}B$$

Watch out for (2)

• Forgetting the xy < 1 condition
  → The Addition Formula — tan⁻¹x ± tan⁻¹y, 2 tan⁻¹x and Three-Term Sums
• Cancelling an inverse outside its range
  → Simplifying by Substitution — x = tan θ, cos θ or cos 2θ

Formulas (3)

• Equations Solved by a Complementary Pair · Replace one of the pair
  $$\cos^{-1}x=\frac{\pi}{2}-\sin^{-1}x \qquad \cot^{-1}x=\frac{\pi}{2}-\tan^{-1}x$$
• Equations Solved by the Addition Formula — and the Roots It Adds · Combine, then take tangents
  $$\tan^{-1}u+\tan^{-1}v=\theta \;\Rightarrow\; \frac{u+v}{1-uv}=\tan\theta\quad(\text{then check }uv<1)$$
• Converting Both Sides to One Function, Substituting, and Checking the Domain · Two conversions and a substitution
  $$\sin(\cot^{-1}u)=\cos(\tan^{-1}u)=\frac{1}{\sqrt{1+u^2}} \qquad x=\tan\theta:\ \sin^{-1}\frac{2x}{1+x^2}=2\theta\ (|x|\le1)$$

Watch out for (3)

• Keeping the root outside the range
  → Equations Solved by a Complementary Pair
• Answering with the number of roots of the quadratic
  → Equations Solved by the Addition Formula — and the Roots It Adds
• Reporting both signs after squaring
  → Converting Both Sides to One Function, Substituting, and Checking the Domain