TRIGONOMETRY STUDY GUIDE — PART 1
Primary Ratios (SOH-CAH-TOA), Reciprocal Functions & Cofunctions
1. INTRODUCTION TO TRIGONOMETRIC RATIOS
Trigonometry focuses on the relationship between the angle measures and side lengths of triangles. For any
acute angle θ in a right-angled triangle, we identify three sides relative to θ: the Hypotenuse (longest side
opposite the right angle), the Opposite side (facing θ), and the Adjacent side (next to θ).
Primary Trigonometric Ratios (SOH-CAH-TOA)
Sine (sin) Cosine (cos) Tangent (tan)
sin(θ) = Opp / Hyp cos(θ) = Adj / Hyp tan(θ) = Opp / Adj
2. RECIPROCAL TRIGONOMETRIC FUNCTIONS
The reciprocal functions are multiplicative inverses of the primary ratios. Flipping each primary ratio yields its
corresponding reciprocal function:
Function Abbr. Ratio Definition Reciprocal Relation Cofunction Identity
Sine sin(θ) Opposite / Hypotenuse 1 / csc(θ) cos(90° − θ)
Cosine cos(θ) Adjacent / Hypotenuse 1 / sec(θ) sin(90° − θ)
Tangent tan(θ) Opposite / Adjacent 1 / cot(θ) cot(90° − θ)
Cosecant csc(θ) Hypotenuse / Opposite 1 / sin(θ) sec(90° − θ)
Secant sec(θ) Hypotenuse / Adjacent 1 / cos(θ) csc(90° − θ)
Cotangent cot(θ) Adjacent / Opposite 1 / tan(θ) tan(90° − θ)
Worked Example 1: Primary & Reciprocal Ratios
Given a right triangle △ABC with legs a = 3, b = 4, and hypotenuse c = 5 (with reference to angle A):
• sin(A) = 3/5 = 0.6 | csc(A) = 5/3 ≈ 1.667
• cos(A) = 4/5 = 0.8 | sec(A) = 5/4 = 1.25
• tan(A) = 3/4 = 0.75 | cot(A) = 4/3 ≈ 1.333
Trigonometry Study Guide Pack — Part 1 Page 1 of 2
Primary Ratios (SOH-CAH-TOA), Reciprocal Functions & Cofunctions
1. INTRODUCTION TO TRIGONOMETRIC RATIOS
Trigonometry focuses on the relationship between the angle measures and side lengths of triangles. For any
acute angle θ in a right-angled triangle, we identify three sides relative to θ: the Hypotenuse (longest side
opposite the right angle), the Opposite side (facing θ), and the Adjacent side (next to θ).
Primary Trigonometric Ratios (SOH-CAH-TOA)
Sine (sin) Cosine (cos) Tangent (tan)
sin(θ) = Opp / Hyp cos(θ) = Adj / Hyp tan(θ) = Opp / Adj
2. RECIPROCAL TRIGONOMETRIC FUNCTIONS
The reciprocal functions are multiplicative inverses of the primary ratios. Flipping each primary ratio yields its
corresponding reciprocal function:
Function Abbr. Ratio Definition Reciprocal Relation Cofunction Identity
Sine sin(θ) Opposite / Hypotenuse 1 / csc(θ) cos(90° − θ)
Cosine cos(θ) Adjacent / Hypotenuse 1 / sec(θ) sin(90° − θ)
Tangent tan(θ) Opposite / Adjacent 1 / cot(θ) cot(90° − θ)
Cosecant csc(θ) Hypotenuse / Opposite 1 / sin(θ) sec(90° − θ)
Secant sec(θ) Hypotenuse / Adjacent 1 / cos(θ) csc(90° − θ)
Cotangent cot(θ) Adjacent / Opposite 1 / tan(θ) tan(90° − θ)
Worked Example 1: Primary & Reciprocal Ratios
Given a right triangle △ABC with legs a = 3, b = 4, and hypotenuse c = 5 (with reference to angle A):
• sin(A) = 3/5 = 0.6 | csc(A) = 5/3 ≈ 1.667
• cos(A) = 4/5 = 0.8 | sec(A) = 5/4 = 1.25
• tan(A) = 3/4 = 0.75 | cot(A) = 4/3 ≈ 1.333
Trigonometry Study Guide Pack — Part 1 Page 1 of 2
