Pythagorean Identities

Pythagorean Identities

How to solve the Pythagorean identities problems: identities, proof, examples, and their solutions.

Identities

sin^2 + cos^2 = 1, tan^2 + 1 = sec^2, 1 + cot^2 = csc^2

sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
These two are the modified identities.

Proof

Pythagorean Identities: Proof of the Identities

Draw a right triangle like this.

Write the central angle as θ.

By the Pythagorean theorem,
x2 + y2 = r2.

Divide both sides by r2:
(x/r)2 + (y/r)2 = 1.

Sine: SOH.
So sin θ = y/r.

Cosine: CAH.
So cos θ = x/r.

Then cos2 θ + sin2 θ = 1.

So sin2 θ + cos2 θ = 1.

Start from sin2 θ + cos2 θ = 1.

Divide both sides by cos2 θ:
sin2 θ/cos2 θ + 1 = 1/cos2 θ.

sin θ/cos θ = tan θ
Quotient identities

And 1/cos θ = sec θ.
Trigonometric ratio - secant

So tan2 θ + 1 = sec2 θ.

Start from sin2 θ + cos2 θ = 1.

Divide both sides by sin2 θ:
1 + cos2 θ/sin2 θ = 1/sin2 θ.

1/sin θ = csc θ.
Trigonometric ratio - cosecant

And cos θ/sin θ = cot θ.
Quotient identities

So 1 + cot2 θ = csc2 θ.

Example 1

Show that the given equation is an identity. (sin theta + cos theta)^2 / (sin theta) = csc theta + 2 cos theta

(sin θ + cos θ)2 = sin2 θ + 2 sin θ cos θ + cos2 θ

Square of a sum

sin2 θ + cos2 θ = 1

Cosecant is the reciprocal of sine.
So 1/sin θ = csc θ.

And cancel sin θ
on both of the numerator and the denominator.

Example 2

Simplify the given expression. (cos^2 theta) / (1 - sin theta)

sin2 θ + cos2 θ = 1

So cos2 θ = 1 - sin2 θ.

Factoring the difference of squares

Example 3

Show that the given equation is an identity. (sin^2 theta)(1 + tan^2 theta) = tan^2 theta

Change 1 + tan2 θ to sec2 θ.

sec2 θ = 1/cos2 θ.

Trigonometric ratio - secant

Quotient identities

Example 4

If sin theta = -3/5 and pi <= theta <= 3*pi/2, find the value of cos theta.

sin θ = -3/5

So (-3/5)2 + sin2 θ = 1.

Then cos2 θ = 16/25.

Draw the axes of the coordinate plane
and write 'all, sin, tan, cos'
like this.

This shows when the trigonometric function is (+):
for quadrant III,
tangent is (+),
and sine and cosine are (-).

πθ ≦ 3π/2
So θ is in quadrant III.

For quadrant III, only tan θ is (+).

So cos θ is (-).

So cos θ = -√16/25.

Simplifying a radical (part 1)

Example 4: Another Solution

If sin theta = -3/5 and pi <= theta <= 3*pi/2, find the value of cos theta.

Let's see how to solve this example differently.

πθ ≦ 3π/2
So θ is in quadrant III.

sin θ = -3/5
And sine: SOH.

So draw a right triangle in quadrant III
whose opposite side is -3
and whose hypotenuse is 5.

This right triangle is a (3, 4, 5) right triangle.

So (blue side) = -4.

Pythagorean triples

The brown angle is the reference angle of ∠θ.

And cosine: CAH.

So cos θ = cos (brown)
= -4/5.

You can see that
you got the same answer.