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Trigonometric Identities
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.
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Trigonometric Identities
What is it?
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.
In simple words
Trigonometric Identities is a core idea in this chapter. Learn the meaning first, then the main parts, then one clear example.
Examples
- sin2 30 + cos2 30 = 1
- If sin = 35, cos = +-45
- Check the quadrant for the sign
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.
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.
Recommended for Class 9
Full school-level explanation
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.
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.
Remember
- divide by hypotenuse^2
Examples
- sin2 30 + cos2 30 = 1
- If sin = 35, cos = +-45
- 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.
Remember
- undefined when the bottom is 0
Examples
- tan 45 = 1
- cos 0 = 1 so sec 0 = 1
- sin 90 = 1 so csc 90 = 1
Remember
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
- identity vs equation
- divide by hypotenuse^2
- undefined when the bottom is 0
Deeper understanding
Go beyond memorising Trigonometric Identities. Connect the definition to reasons, relationships, exceptions, and exam wording.
Exam focus
Quick revision
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
- identity vs equation
- PYTHAGOREAN IDENTITY: From a unit circle or a right triangle, opposite^2 + adjacent^2 = hypotenuse^2. Divide by hypotenuse
- RECIPROCAL AND QUOTIENT: csc theta = 1/sin theta (sin not 0). sec = 1/cos. cot = cos/sin = 1/tan. These are definitions, not
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.
For long answers on Trigonometric Identities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
- identity vs equation
Important formulas
- sin2 + cos2 = 1
- tan = sin/cos
- reciprocal ratios
Common exam traps
- 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.
- 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.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- 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' sin^2 + cos^2 = 1 for theta.
- 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' sin^2 + cos^2 = 1 for theta.
- 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' sin^2 + cos^2 = 1 for theta.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sin2 + cos2 = 1
tan = sin/cos
reciprocal ratios
identity vs equation
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,
sin2 + cos2 = 1
- 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,
- Formula
- sin2 + cos2 = 1
- Example
- sin2 30 + cos2 30 = 1
- Critical warning
- 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 +
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Exam Practice
Define
Q1. Define Trigonometric Identities.
Explain
Q2. Explain: PYTHAGOREAN IDENTITY.
Example
Q3. Give one clear example of Trigonometric Identities and say why it fits.
Mistake
Q4. State one common mistake students make about Trigonometric Identities.
Exam note
Q5. Write a short exam note on Trigonometric Identities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.