Further Symmetry Properties and the Pythagorean Identity Revision Notes for VCE SSCE Mathematical Methods

Further Symmetry Properties and the Pythagorean Identity

Complementary relationships

Complementary relationships, also known as cofunction identities, show how sine and cosine are related when angles differ by $\frac{\pi}{2}$ radians (90°). These relationships are powerful tools for simplifying trigonometric expressions.

Angles of the form $\frac{\pi}{2} - \theta$

Examining the unit circle, we can establish the following relationships:

\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

These identities tell us that the sine of an angle equals the cosine of its complement, and vice versa. This is why sine and cosine are called cofunctions.

infoNote

Think of it this way: when angles are complementary (add up to $\frac{\pi}{2}$ ), sine and cosine "swap" their roles. This relationship is fundamental to understanding how these functions relate to each other.

Angles of the form $\frac{\pi}{2} + \theta$

Similarly, for angles that exceed $\frac{\pi}{2}$ by an amount $\theta$ :

\sin\left(\frac{\pi}{2} + \theta\right) = \cos \theta

\cos\left(\frac{\pi}{2} + \theta\right) = -\sin \theta

Notice that the sine relationship remains the same, but the cosine relationship introduces a negative sign.

chatImportant

The negative sign in $\cos\left(\frac{\pi}{2} + \theta\right) = -\sin \theta$ is crucial! This is due to the point's position on the unit circle moving into the second quadrant where cosine values are negative.

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Worked Example: Using Complementary Relationships

Problem: If $\sin \theta = 0.3$ and $\cos \psi = 0.8$ , find the value of:

a) $\sin\left(\frac{\pi}{2} - \psi\right)$

b) $\cos\left(\frac{\pi}{2} + \theta\right)$

Solution:

Part a:

Using the complementary relationship:

\sin\left(\frac{\pi}{2} - \psi\right) = \cos \psi = 0.8

Part b:

Using the complementary relationship:

\cos\left(\frac{\pi}{2} + \theta\right) = -\sin \theta = -0.3

The Pythagorean identity

The Pythagorean identity is one of the most important relationships in trigonometry. It connects the sine and cosine of any angle through the equation:

\cos^2\theta + \sin^2\theta = 1

chatImportant

This identity holds true for all values of $\theta$ , making it one of the most versatile tools in trigonometry.

Derivation of the Pythagorean identity

We can derive this identity by considering a point $P(\theta)$ located on the unit circle.

Let $P(\theta)$ be a point on the unit circle, where the circle has radius 1. The coordinates of this point are $(\cos \theta, \sin \theta)$ .

Using Pythagoras' theorem on the right triangle formed:

OP^2 = OM^2 + MP^2

Since the radius $OP = 1$ , and the horizontal distance $OM = \cos \theta$ and vertical distance $MP = \sin \theta$ :

1 = (\cos \theta)^2 + (\sin \theta)^2

We can write $(\cos \theta)^2$ as $\cos^2 \theta$ and $(\sin \theta)^2$ as $\sin^2 \theta$ . Therefore:

\cos^2\theta + \sin^2\theta = 1

This is the Pythagorean identity, derived directly from the geometry of the unit circle.

infoNote

The beauty of this derivation is that it shows how Pythagoras' theorem directly translates to trigonometry. On the unit circle, the equation $x^2 + y^2 = 1$ becomes $\cos^2\theta + \sin^2\theta = 1$ when we recognize that the coordinates are $(\cos \theta, \sin \theta)$ .

Using the Pythagorean identity

The Pythagorean identity is extremely useful for finding one trigonometric ratio when another is known, especially when combined with quadrant information.

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Worked Example: Using the Pythagorean Identity to Find Trigonometric Values

Problem: Given that $\sin x = \frac{3}{5}$ and $\frac{\pi}{2} < x < \pi$ , find:

a) $\cos x$

b) $\tan x$

Solution:

Part a:

Substitute $\sin x = \frac{3}{5}$ into the Pythagorean identity:

\cos^2 x + \sin^2 x = 1

\cos^2 x + \frac{9}{25} = 1

\cos^2 x = 1 - \frac{9}{25}

\cos^2 x = \frac{16}{25}

Therefore $\cos x = \pm\frac{4}{5}$ .

Since $x$ is in the second quadrant (where $\frac{\pi}{2} < x < \pi$ ), and cosine is negative in the second quadrant:

\cos x = -\frac{4}{5}

Part b:

Using the result from part a:

\tan x = \frac{\sin x}{\cos x}

Exam tip: Always consider which quadrant the angle is in when determining the sign of your answer. This is crucial for selecting the correct value when taking square roots.

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Key Points to Remember:

Complementary relationships show that sine and cosine swap when the angle changes by $\frac{\pi}{2}$ : $\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$ and $\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$
When adding $\frac{\pi}{2}$ to an angle: $\sin\left(\frac{\pi}{2} + \theta\right) = \cos \theta$ but $\cos\left(\frac{\pi}{2} + \theta\right) = -\sin \theta$ (note the negative sign)
The Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$ is valid for all values of $\theta$ and is derived from Pythagoras' theorem applied to the unit circle
When using the Pythagorean identity to find a trigonometric ratio, always check the quadrant to determine the correct sign
The Pythagorean identity can be rearranged to find either $\sin^2\theta = 1 - \cos^2\theta$ or $\cos^2\theta = 1 - \sin^2\theta$

Further Symmetry Properties and the Pythagorean Identity (VCE SSCE Mathematical Methods): Revision Notes