MathematicsChapter 6: Trigonometry and Bearing
Trigonometric Identities
An identity is true for all allowed angles, not just one triangle you measured.
Definition
An identity is true for all allowed angles, not just one triangle you measured. Class 9 needs sin^2 theta + cos^2 theta = 1, tan theta = sin theta / cos theta, and the reciprocal links: csc = 1/sin, sec = 1/cos, cot = 1/tan.
Mathematical form
theta = 1
theta = sin theta / cos theta
csc = 1/sin
sec = 1/cos
Relationship
- sin2 + cos2 = 1
- tan = sin/cos
Overview
Two jobs on a paper:
Pythagorean Identity
From a unit circle or a right triangle, opposite^2 + adjacent^2 = hypotenuse^2. Divide by hypotenuse^2 to get sin^2 + cos^2 = 1. Then 1 + tan^2 = sec^2 follows by dividing by cos^2.
Examples sin^2 30 + cos^2 30 = 1 · If sin = 3/5, cos = +-4/5 · Check the quadrant for the sign
Reciprocal and Quotient
csc theta = 1/sin theta (sin not 0). sec = 1/cos. cot = cos/sin = 1/tan. These are definitions, not new triangles. If sin theta = 0, csc is undefined.
Examples tan 45 = 1 · cos 0 = 1 so sec 0 = 1 · sin 90 = 1 so csc 90 = 1
In simple words
An equation such as 2 sin theta = 1 is to be solved. An identity is to be shown true in general. Do not 'solve' sin^2 + cos^2 = 1 for theta.
Key points
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
- divide by hypotenuse^2
- undefined when the bottom is 0
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Trigonometric Identities
Sticky Notes · Quick Revision
Core Idea
- An identity is true for all allowed angles, not just one triangle you measured.
Key Points
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
Simple Check
- Explain trigonometric identities in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- An equation such as 2 sin theta = 1 is to be solved. An identity is to be shown true in general. Do not 'solve' sin2 + cos2 = 1 for theta.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
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