In a right triangle, trigonometric ratios connect an angle with the ratios of two sides. For any angle θ (theta), there are six basic ratios: sine, cosine, tangent, cosecant, secant, and cotangent
💡Key Definitions
Consider a right triangle with angle θ. The three sides are: Hypotenuse (longest side, opposite to 90°), Opposite (side facing angle θ), and Adjacent (side next to angle θ)
→Basic Ratios
sin θ = Opposite/Hypotenuse
cos θ = Adjacent/Hypotenuse
tan θ = Opposite/Adjacent
cosec θ = 1/sin θ = Hypotenuse/Opposite
sec θ = 1/cos θ = Hypotenuse/Adjacent
cot θ = 1/tan θ = Adjacent/Opposite
Fundamental Identity: sin²θ + cos²θ = 1 (Most important for SSC CGL)
Other Identities: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
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Exam Patterns
What examiners ask — read before attempting PYQs
✏️Worked Example 2
1
Replace cosec θ = 1/sin θ and sec θ = 1/cos θ
2
(sin θ × 1/sin θ) + (cos θ × 1/cos θ)
3
1 + 1 = 2
Answer: 2
Shortcut #3 - Reciprocal Recognition:
Instantly recognize reciprocal pairs: sin-cosec, cos-sec, tan-cot. Their product always equals 1.
Common Mistake #1: Students often confuse opposite and adjacent sides when angle position changes. Always identify the angle first, then mark opposite and adjacent accordingly. Many students lose marks by mixing up sin and cos definitions when the triangle orientation changes. Practice identifying sides relative to the given angle, not the triangle's position on paper
💡Practical Exam Tip
In multiple choice questions, if you get values like sin θ = 4/3 or cos θ = 6/5, immediately mark it wrong. Sine and cosine values cannot exceed 1. This elimination technique saves precious exam time.
If cos θ = 7/25 and θ is acute, what is the value of sin θ + tan θ?
Practice 3medium
If sin θ = 3/5 and θ is an acute angle, find the value of tan θ.
Practice 4medium
If sec θ - tan θ = 2/5, find the value of sec θ + tan θ.
Practice 5medium
A ladder leans against a wall. The ladder is 13 metres long and makes an angle of 67.38° with the ground. Approximately, how high does the ladder reach on the wall? (Use sin 67.38° ≈ 0.92)
60-Second Revision — Basic Trig Ratios
Remember: sin²θ + cos²θ = 1 is the most tested identity
Formula: Standard angles - sin 30°=1/2, sin 60°=√3/2, sin 45°=1/√2
Trap: Never accept sin θ or cos θ values greater than 1 in any answer choice
Quick tip: cosec, sec, cot are just reciprocals - flip the fraction
Check: Opposite side is always relative to the given angle, not triangle orientation
Speed trick: Use complementary angle property - sin θ = cos(90°-θ)
Final check: In MCQs, eliminate impossible ratio values first