Trigonometric Identities Revision Notes for Grade 11 NSC Matric Mathematics

Trigonometric Identities

What are trigonometric identities?

A trigonometric identity is a mathematical statement that shows two different trigonometric expressions are equal for all valid values of the variable. These identities are incredibly useful tools that help us simplify complex trigonometric expressions and solve equations more efficiently.

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The main purpose of trigonometric identities is to transform expressions so they contain only sine and cosine ratios, making calculations much simpler. This is why the strategy "convert everything to sine and cosine" is so effective!

The quotient identity

Understanding the quotient identity

The quotient identity shows us the relationship between tangent, sine, and cosine functions. We know from basic trigonometry that:

$\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$

However, we can also express tangent in terms of sine and cosine ratios.

Using coordinate geometry and the definitions of trigonometric ratios, we can derive this relationship. Consider a point (x, y) on the coordinate plane where the distance from the origin is r.

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Deriving the Quotient Identity

From the coordinate definitions:

$\sin \theta = \frac{y}{r}$
$\cos \theta = \frac{x}{r}$

Starting with the basic definition of tangent: $\tan \theta = \frac{y}{x}$

We can multiply both the numerator and denominator by $\frac{r}{r}$ : $\tan \theta = \frac{y}{x} \times \frac{r}{r} = \frac{y \cdot r}{x \cdot r} = \frac{y}{r} \div \frac{x}{r}$

This gives us: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

This is the quotient identity:

$\boxed{\tan \theta = \frac{\sin \theta}{\cos \theta}}$

Restrictions and undefined values

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It's crucial to remember that $\tan \theta$ is undefined when $\cos \theta = 0$ . This occurs when $\theta = k \times 90°$ where k is an odd integer (e.g., 90°, 270°, 450°, etc.).

Always check for these restrictions when working with tangent functions!

The graph of the tangent function shows these restrictions as vertical asymptotes, where the function approaches positive or negative infinity.

The square identity (Pythagorean identity)

Deriving the square identity

The square identity is one of the most fundamental relationships in trigonometry. It comes directly from the Pythagorean theorem.

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Deriving the Square Identity from the Pythagorean Theorem

Consider a point (x, y) on the coordinate plane, where r is the distance from the origin to the point. Using the Pythagorean theorem: $x^2 + y^2 = r^2$

Now, we know that:

$\sin \theta = \frac{y}{r}$ , so $y = r \sin \theta$
$\cos \theta = \frac{x}{r}$ , so $x = r \cos \theta$

Substituting these into the Pythagorean theorem:

$(r \cos \theta)^2 + (r \sin \theta)^2 = r^2$
$r^2 \cos^2 \theta + r^2 \sin^2 \theta = r^2$
$r^2(\cos^2 \theta + \sin^2 \theta) = r^2$

Dividing both sides by $r^2$ :

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

This gives us the square identity (also called the Pythagorean identity):

$\boxed{\sin^2 \theta + \cos^2 \theta = 1}$

Alternative forms of the square identity

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The square identity can be rearranged into different useful forms:

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

These alternative forms are particularly useful when simplifying expressions or solving equations.

Tips for proving identities

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Strategic Approaches for Proving Trigonometric Identities

When working with trigonometric identities, follow these proven strategies:

Convert everything to sine and cosine - This is often the most reliable method
Choose one side to simplify - Usually work with the more complicated side and show it equals the simpler side
Work with the more complex side first - This typically leads to easier simplification
Consider both sides - Sometimes you need to simplify both sides to show they're equal
Look for square root signs - These often indicate you should use the square identity
Use algebraic manipulation - You can add terms strategically to make simplification easier, such as:
- Replace 1 with $\sin^2 \theta + \cos^2 \theta$
- Multiply by suitable fractions like $\frac{1 + \sin \theta}{1 + \sin \theta}$

Worked examples

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Worked Example 1: Simplifying expressions

Question: Simplify $\tan^2 \theta \times \cos^2 \theta$

Solution:

Step 1: Convert to sine and cosine using the quotient identity

$\tan^2 \theta \times \cos^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 \times \cos^2 \theta$

Step 2: Simplify the expression

$= \frac{\sin^2 \theta}{\cos^2 \theta} \times \cos^2 \theta = \sin^2 \theta$

Therefore: $\tan^2 \theta \times \cos^2 \theta = \sin^2 \theta$

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Worked Example 2: Simplifying complex fractions

Question: Simplify $\frac{1}{\cos^2 \theta} - \tan^2 \theta$

Solution:

Step 1: Convert tangent to sine and cosine

$\frac{1}{\cos^2 \theta} - \tan^2 \theta = \frac{1}{\cos^2 \theta} - \left(\frac{\sin \theta}{\cos \theta}\right)^2$

Step 2: Express with common denominator

$= \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1 - \sin^2 \theta}{\cos^2 \theta}$

Step 3: Use the square identity ( $1 - \sin^2 \theta = \cos^2 \theta$ )

$= \frac{\cos^2 \theta}{\cos^2 \theta} = 1$

Therefore: $\frac{1}{\cos^2 \theta} - \tan^2 \theta = 1$

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Worked Example 3: Proving an identity

Question: Prove that $\frac{1 - \sin \alpha}{\cos \alpha} = \frac{\cos \alpha}{1 + \sin \alpha}$

Solution:

Step 1: Note the restrictions

For this identity to be valid, we need $\cos \alpha \neq 0$ and $\sin \alpha \neq -1$ .

Step 2: Simplify the left-hand side

We'll multiply the numerator and denominator by $(1 + \sin \alpha)$ :

$\text{LHS} = \frac{1 - \sin \alpha}{\cos \alpha} \times \frac{1 + \sin \alpha}{1 + \sin \alpha}$

Step 3: Apply the difference of squares

$= \frac{(1 - \sin \alpha)(1 + \sin \alpha)}{\cos \alpha(1 + \sin \alpha)} = \frac{1 - \sin^2 \alpha}{\cos \alpha(1 + \sin \alpha)}$

Step 4: Use the square identity ( $1 - \sin^2 \alpha = \cos^2 \alpha$ )

$= \frac{\cos^2 \alpha}{\cos \alpha(1 + \sin \alpha)} = \frac{\cos \alpha}{1 + \sin \alpha}$

This equals the right-hand side, so the identity is proven.

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Key Points to Remember:

Trigonometric identities are equations that are true for all valid values of the angle
The quotient identity is $\tan \theta = \frac{\sin \theta}{\cos \theta}$ , but remember that tangent is undefined when cosine equals zero
The square identity is $\sin^2 \theta + \cos^2 \theta = 1$ , which comes from the Pythagorean theorem
When proving identities, convert everything to sine and cosine, work with the more complex side first, and use algebraic manipulation strategically
Always check for restrictions
- identify values where denominators become zero or expressions become undefined

Trigonometric Identities (Grade 11 NSC Matric Mathematics): Revision Notes