Trigonometry: SSC CGL Quantitative Aptitude

Formulas (2)

• The six trigonometric ratios · The six ratios
  $$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{\sin\theta}{\cos\theta} \qquad \text{cosec}\,\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$
• Converting degrees and radians · Degree and radian conversion
  $$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi} \qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180^\circ}$$

Reference tables (1)

Watch out for (8)

• Pairing cosec with cos
  → The six trigonometric ratios
• Turning tan the wrong way up
  → The six trigonometric ratios
• Swapping sin 60° and cos 60°
  → Values of the six ratios at 0°, 30°, 45°, 60° and 90°
• Treating tan 90° as a number
  → Values of the six ratios at 0°, 30°, 45°, 60° and 90°
• Reading the angle off the wrong row
  → Reading the angle from a value
• Taking the angle as the answer
  → Reading the angle from a value
• Multiplying by the wrong factor
  → Converting degrees and radians
• Rounding 1 radian to 60°
  → Converting degrees and radians

Formulas (3)

• Ratios from the sides of a right triangle · Sides of a right triangle
  $$a^2 + b^2 = c^2 \qquad \text{opposite} = \text{hypotenuse} \times \sin\theta \qquad \text{adjacent} = \text{hypotenuse} \times \cos\theta$$
• One ratio gives the other five · From tan θ = a/b
  $$\sin\theta = \frac{a}{\sqrt{a^2 + b^2}} \qquad \cos\theta = \frac{b}{\sqrt{a^2 + b^2}}$$
• Dividing by cos θ when tan θ is given · Divide through by cos θ
  $$\frac{a\sin\theta + b\cos\theta}{c\sin\theta + d\cos\theta} = \frac{a\tan\theta + b}{c\tan\theta + d}$$

Watch out for (6)

• Using the longest printed number as a leg
  → Ratios from the sides of a right triangle
• Opposite and adjacent belong to one angle
  → Ratios from the sides of a right triangle
• Using the triangle outside the first quadrant
  → One ratio gives the other five
• Rejecting a correct answer that looks different
  → One ratio gives the other five
• Dividing when a constant term is present
  → Dividing by cos θ when tan θ is given
• Getting tan after dividing by sin
  → Dividing by cos θ when tan θ is given

Formulas (3)

• The three Pythagorean identities · Pythagorean identities
  $$\sin^2\theta + \cos^2\theta = 1 \qquad 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \text{cosec}^2\theta$$
• Rewrite in sin and cos, then simplify · Factor patterns with a² + b² = 1
  $$a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2) \qquad (1 - \cos\theta)(1 + \cos\theta) = \sin^2\theta$$
• Eliminating θ from two equations · Removing θ
  $$x = a\cos\theta,\ y = b\sin\theta \Rightarrow \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \qquad x = a\sec\theta,\ y = b\tan\theta \Rightarrow \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \qquad (a\cos\theta + b\sin\theta)^2 + (a\sin\theta - b\cos\theta)^2 = a^2 + b^2$$

Watch out for (7)

• A minus sign that is not an identity
  → The three Pythagorean identities
• Moving 1 across with the wrong sign
  → The three Pythagorean identities
• One passing angle is not a proof
  → The three Pythagorean identities
• Cancelling across a plus sign
  → Rewrite in sin and cos, then simplify
• A test angle where two options agree
  → Rewrite in sin and cos, then simplify
• Adding where the identity subtracts
  → Eliminating θ from two equations
• Expanding without checking the cross terms
  → Eliminating θ from two equations

Formulas (3)

• The co-function rule for complementary angles · Co-function rule
  $$\sin(90^\circ - \theta) = \cos\theta \qquad \tan(90^\circ - \theta) = \cot\theta \qquad \sec(90^\circ - \theta) = \text{cosec}\,\theta$$
• Angles that add up to 90° · Partners that make 90°
  $$\sin A = \cos B \ \text{or} \ \tan A = \cot B \ (A, B \text{ acute}) \Rightarrow A + B = 90^\circ$$
• Allied angles, quadrant signs and large angles · Allied angles
  $$\sin(180^\circ - \theta) = \sin\theta \qquad \cos(180^\circ - \theta) = -\cos\theta \qquad \tan(180^\circ + \theta) = \tan\theta \qquad \cos(360^\circ - \theta) = \cos\theta \qquad \sin(-\theta) = -\sin\theta$$

Watch out for (7)

• Pairing angles that add to 180°
  → The co-function rule for complementary angles
• Pairing a ratio with itself
  → The co-function rule for complementary angles
• Setting the angles equal
  → Angles that add up to 90°
• Using a non-standard answer as a sign of error
  → Angles that add up to 90°
• Dropping the sign in the second quadrant
  → Allied angles, quadrant signs and large angles
• Taking away 180° from a sine
  → Allied angles, quadrant signs and large angles
• Putting a minus on cos of a negative angle
  → Allied angles, quadrant signs and large angles

Formulas (4)

• sec θ + tan θ and sec θ − tan θ multiply to 1 · The reciprocal pairs
  $$(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1 \qquad (\text{cosec}\,\theta + \cot\theta)(\text{cosec}\,\theta - \cot\theta) = 1 \qquad \sec\theta + \tan\theta = p \Rightarrow \sin\theta = \frac{p^2 - 1}{p^2 + 1} \qquad \sqrt{\frac{1 + \sin\theta}{1 - \sin\theta}} = \sec\theta + \tan\theta$$
• Squaring a given sum · Squared sums
  $$(\sin\theta \pm \cos\theta)^2 = 1 \pm 2\sin\theta\cos\theta \qquad (\sin\theta + \cos\theta)^2 + (\sin\theta - \cos\theta)^2 = 2 \qquad x + \frac{1}{x} = k \Rightarrow x^2 + \frac{1}{x^2} = k^2 - 2 \qquad \tan\theta + \cot\theta = \frac{1}{\sin\theta\cos\theta}$$
• sin⁴θ + cos⁴θ and sin⁶θ + cos⁶θ · Higher powers of sin and cos
  $$\sin^4\theta + \cos^4\theta = 1 - 2\sin^2\theta\cos^2\theta \qquad \sin^6\theta + \cos^6\theta = 1 - 3\sin^2\theta\cos^2\theta \qquad \sin^4\theta - \cos^4\theta = \sin^2\theta - \cos^2\theta$$
• Substitution chains such as cos θ + cos²θ = 1 · The two chains
  $$\cos\theta + \cos^2\theta = 1 \Rightarrow \cos\theta = \sin^2\theta \qquad \sin\theta + \sin^2\theta = 1 \Rightarrow \sin\theta = \cos^2\theta$$

Watch out for (7)

• Taking the partner as p − 1 or −p
  → sec θ + tan θ and sec θ − tan θ multiply to 1
• Swapping the sum and the difference
  → sec θ + tan θ and sec θ − tan θ multiply to 1
• Forgetting the 2 in the cross term
  → Squaring a given sum
• Answering a given that cannot happen
  → Squaring a given sum
• Writing sin⁴θ + cos⁴θ = 1
  → sin⁴θ + cos⁴θ and sin⁶θ + cos⁶θ
• Using θ = 0° when the options contain θ
  → sin⁴θ + cos⁴θ and sin⁶θ + cos⁶θ
• Hunting for the angle
  → Substitution chains such as cos θ + cos²θ = 1

Formulas (3)

• The addition and subtraction formulas · Compound angles
  $$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B \qquad \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B \qquad \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$
• Double, half and triple angle formulas · Multiple angles
  $$\sin 2A = 2\sin A\cos A = \frac{2\tan A}{1 + \tan^2 A} \qquad \cos 2A = 1 - 2\sin^2 A = 2\cos^2 A - 1 = \frac{1 - \tan^2 A}{1 + \tan^2 A} \qquad \tan\frac{A}{2} = \frac{\sin A}{1 + \cos A} \qquad \sin 3A = 3\sin A - 4\sin^3 A \qquad \cos 3A = 4\cos^3 A - 3\cos A$$
• Products and sums of sines and cosines · Product and sum formulas
  $$2\sin A\sin B = \cos(A - B) - \cos(A + B) \qquad 2\cos A\cos B = \cos(A - B) + \cos(A + B) \qquad 2\sin A\cos B = \sin(A + B) + \sin(A - B) \qquad \sin C + \sin D = 2\sin\tfrac{C + D}{2}\cos\tfrac{C - D}{2} \qquad \cos C + \cos D = 2\cos\tfrac{C + D}{2}\cos\tfrac{C - D}{2}$$

Watch out for (5)

• Keeping the sign in cos(A + B)
  → The addition and subtraction formulas
• Splitting tan over a sum
  → The addition and subtraction formulas
• Doubling the ratio instead of the angle
  → Double, half and triple angle formulas
• Turning the half-angle fraction upside down
  → Double, half and triple angle formulas
• Reversing the order in 2 sin A sin B
  → Products and sums of sines and cosines

Formulas (2)

• Reduce to one ratio, then read the angle · Mixed squares of sin and cos
  $$a\sin^2\theta + b\cos^2\theta = c \Rightarrow \sin^2\theta = \frac{c - b}{a - b}$$
• Greatest and least values of a sin θ + b cos θ and of powers · Range of a sin θ + b cos θ
  $$-\sqrt{a^2 + b^2} \le a\sin\theta + b\cos\theta \le \sqrt{a^2 + b^2}$$

Watch out for (4)

• Keeping an impossible root
  → Reduce to one ratio, then read the angle
• Cancelling a factor that can be zero
  → Reduce to one ratio, then read the angle
• Adding the coefficients
  → Greatest and least values of a sin θ + b cos θ and of powers
• Forgetting the least value is negative
  → Greatest and least values of a sin θ + b cos θ and of powers