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Thought-Process to Discover Knowledge

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mathsTrigonometryTrigonometric Identities and Complementary Angles

### Pythagorean Trigonometric Identities

The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"

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For the triangle given in figure, sin theta = (PQ)/(OQ)

For the triangle given in figure, cos theta = (OP)/(OQ)

The Pythagoras theorem is given as

OP^2 + PQ^2 =OQ^2

Or equivalently

((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1

For a right angled triangle, the Pythagoras theorem is given as

(text(opposite))^2 + (text(adjacent))^2 = (text(hypotenuse))^2
OP^2 + PQ^2 =OQ^2

If this equation is divided by (text(hypotenuse))^2 or OQ^2
((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1

Or equivalently

sin^2 theta + cos^2 theta = 1

Pythagorean Theorem states that in a right angled triangle, square of hypotenuse equals sum of squares of two arms. The trigonometric ratios are defined for right angled triangles. The relationships between trigonometric ratios per Pythagorean theorem are called "Pythagorean Trigonometric Identities".

sin^2 theta + cos^2 theta = 1

It is noted that the result is true for any value of theta. That is, if theta = 27, then sin^2 27^@ + cos^2 27^@ = 1

The word "identity" means equality of two expressions; left and right hand side are identical.

In the Pythagorean Trigonometric Identity sin^2 theta + cos^2 theta = 1, it is stated that left hand side sin^2 theta + cos^2 theta equals the right hand side 1.

For a right angled triangle,

sin^2 theta + cos^2 theta = 1

If this equation is divided by sin^2 theta, the following identity is derived

1+cot^2 theta = csc^2 theta

1+cot^2 theta = csc^2 theta

For a right angled triangle,

sin^2 theta + cos^2 theta = 1

If this equation is divided by cos^2 theta, the following identity is derived.

tan^2 theta + 1 = sec^2 theta

tan^2 theta + 1 = sec^2 theta

Pythagorean Trigonometric Identities:

For any theta,

sin^2 theta + cos^2 theta = 1

1+cot^2 theta = csc^2 theta

tan^2 theta + 1 = sec^2 theta

Note that this need not me memorized, connect these to the Pythogoras theorem and quickly derive when required.

Solved Exercise Problem:

What is the value of 1-sin^2 theta?

• cos^2 theta
• cos^2 theta
• the question is incomplete to find an answer

The answer is 'cos^2 theta'.

This is derived from the Pythagorean trigonometric identity sin^2 theta + cos^2 theta = 1

Solved Exercise Problem:

What is the value of sec^2 theta - 1?

• tan^2 theta
• tan^2 theta
• cot^2 theta

The answer is 'tan^2 theta'.

This is derived from the Pythagorean trigonometric identity tan^2 theta + 1 = sec^2 theta

Solved Exercise Problem:

What is the value of csc^2 theta-cot^2 theta?

• cos^2 theta
• 1
• 1

The answer is '1'.

This is derived from the Pythagorean trigonometric identity 1 + cot^2 theta = csc^2 theta

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