Trigonometry - II — MHT-CET Maths

Formulas (2)

• The Compound-Angle Formulas and a sin x + b cos x · Compound angles
  $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \qquad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B \qquad \tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$$
• Fixed Angle Sums and the Tangent Formula · Clearing the fraction
  $$\tan(A+B)=k \iff \tan A+\tan B = k\,(1-\tan A\tan B)$$

Watch out for (5)

• Keeping the plus sign in cos(A + B)
  → The Compound-Angle Formulas and a sin x + b cos x
• Taking a + b as the maximum of a sin x + b cos x
  → The Compound-Angle Formulas and a sin x + b cos x
• Forgetting that cot is the reciprocal
  → The Compound-Angle Formulas and a sin x + b cos x
• Reading 225° as a new case
  → Fixed Angle Sums and the Tangent Formula
• Dropping a sign in the triple rearrangement
  → Fixed Angle Sums and the Tangent Formula

Formulas (2)

• Double and Half Angles, and the Sign of a Half Angle · Double and half angles
  $$\cos 2A = 2\cos^2 A - 1 = 1 - 2\sin^2 A \qquad \cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}} \qquad \tan\frac{x}{2}=\frac{1-\cos x}{\sin x}$$
• Triple Angles, Standard Values and Pairing · Triple angles
  $$\sin 3A = 3\sin A - 4\sin^3 A \qquad \cos 3A = 4\cos^3 A - 3\cos A \qquad \tan 3A=\frac{3\tan A-\tan^3 A}{1-3\tan^2 A}$$

Watch out for (5)

• Taking the sign from the quadrant of x
  → Double and Half Angles, and the Sign of a Half Angle
• Using the wrong form of cos 2A
  → Double and Half Angles, and the Sign of a Half Angle
• The negative root of tan(π/8)
  → Double and Half Angles, and the Sign of a Half Angle
• Multiplying out four brackets
  → Triple Angles, Standard Values and Pairing
• Mixing up sin 18° and cos 36°
  → Triple Angles, Standard Values and Pairing

Formulas (2)

• Sums to Products, and Ratio Conditions · Sum to product
  $$\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2} \qquad \cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}$$
• Products to Sums, and the Two Square-Difference Identities · Product to sum
  $$2\cos A\cos B=\cos(A-B)+\cos(A+B) \qquad \cos^2 A-\sin^2 B=\cos(A+B)\cos(A-B)$$

Watch out for (4)

• The minus sign in cos C − cos D
  → Sums to Products, and Ratio Conditions
• Inverting the ratio in componendo and dividendo
  → Sums to Products, and Ratio Conditions
• Using the sine pair for a cosine-minus-sine
  → Products to Sums, and the Two Square-Difference Identities
• Using 18° where the value needs 36°
  → Products to Sums, and the Two Square-Difference Identities