Trigonometry: Ratios, Identities and Heights
Trigonometry in the SSC course is confined to right triangles: the six ratios, the standard-angle table, three identities, complementary angles, and applications to heights and distances.
Definitions
For an acute angle θ in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent = sin θ/cos θ. cosec θ, sec θ and cot θ are the reciprocals of sin θ, cos θ and tan θ respectively.
Standard angles
sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1; the cosine row is the same list reversed. tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, and tan 90° is undefined.
Identities and complements
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ. Complementary angles give sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.
Heights and distances
Draw the figure first. The angle of elevation is measured upward from the horizontal at the observer, the angle of depression downward from the horizontal at the observer. Use tan when a height and a horizontal distance are connected, and sin or cos when the line of sight itself is given.
Worked example
If tan θ = 3/4 and θ is acute, find sin θ + cos θ. Take opposite = 3, adjacent = 4, so hypotenuse = √(9 + 16) = 5. Then sin θ = 3/5, cos θ = 4/5, and sin θ + cos θ = 7/5.
Traps that cost marks
Mixing up the 30° and 60° values, writing sin²θ = 1 − cos θ instead of 1 − cos²θ, measuring the angle of depression from the vertical, and taking the negative root when θ is stated to be acute are the standard errors. Calculators are not permitted, so the standard-angle values must be memorised.