📐 Level 1 - Topic 9: Basic Trigonometric Identities 🎭

1) What are Trigonometric Identities? - Unveiling the "True" Equations 🎭

In mathematics, an identity is an equation that is always true, no matter what values are substituted for the variables (for which the expressions are defined). In trigonometry, trigonometric identities are equations that are true for all angles for which the trigonometric functions in the equation are defined.

A Trigonometric Identity is an equation involving trigonometric functions that is true for all angles for which the functions are defined.

Why are Trigonometric Identities Important?

Simplification: Identities help simplify complex trigonometric expressions, making them easier to work with in calculations and proofs.
Solving Equations: They are essential for solving trigonometric equations, allowing us to rewrite equations in a more manageable form.
Proofs and Derivations: Identities are the building blocks for proving other trigonometric relationships and deriving new formulas.
Mathematical Structure: They reveal fundamental relationships and structure within trigonometry.

We've already encountered one very important trigonometric identity: the Pythagorean Identity. In this topic, we'll explore a set of fundamental identities that are essential in trigonometry.

2) Reciprocal Identities - A Quick Recap 🔄

We've already learned about reciprocal trigonometric functions in Topic 7. The relationships defining them are actually our first set of trigonometric identities:

Reciprocal Identities:

\( \csc(\theta) = \frac{1}{\sin(\theta)} \quad \Leftrightarrow \quad \sin(\theta) = \frac{1}{\csc(\theta)} \)

\( \sec(\theta) = \frac{1}{\cos(\theta)} \quad \Leftrightarrow \quad \cos(\theta) = \frac{1}{\sec(\theta)} \)

\( \cot(\theta) = \frac{1}{\tan(\theta)} \quad \Leftrightarrow \quad \tan(\theta) = \frac{1}{\cot(\theta)} \)

These identities are true for all angles \( \theta \) for which the denominators are not zero. For example, \( \csc(\theta) = \frac{1}{\sin(\theta)} \) is true as long as \( \sin(\theta) \neq 0 \).

3) Ratio Identities - Tangent and Cotangent in Terms of Sine and Cosine ➗

We know that tangent and cotangent are related to sine and cosine. These relationships give us another set of fundamental identities:

Ratio Identities:

\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

These identities are also true for all angles where the denominators are non-zero. Specifically, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) is true for \( \cos(\theta) \neq 0 \), and \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) is true for \( \sin(\theta) \neq 0 \). Notice that \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{\sin(\theta)}{\cos(\theta)}} = \frac{\cos(\theta)}{\sin(\theta)} \), so the ratio identity for cotangent is consistent with its reciprocal identity.

4) Pythagorean Identities - Expanding on \( \sin^2(\theta) + \cos^2(\theta) = 1 \) 💪

We've already learned the primary Pythagorean Identity in Topic 8: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). But we can derive two more useful identities from this one.

Starting with: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

Identity 1: Divide by \( \cos^2(\theta) \) (assuming \( \cos(\theta) \neq 0 \)):

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

\( \left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2 + 1 = \left(\frac{1}{\cos(\theta)}\right)^2 \)

Using ratio identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and reciprocal identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we get:

\( \tan^2(\theta) + 1 = \sec^2(\theta) \) or \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

Identity 2: Divide by \( \sin^2(\theta) \) (assuming \( \sin(\theta) \neq 0 \)):

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

\( 1 + \left(\frac{\cos(\theta)}{\sin(\theta)}\right)^2 = \left(\frac{1}{\sin(\theta)}\right)^2 \)

Using ratio identity \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and reciprocal identity \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we get:

\( 1 + \cot^2(\theta) = \csc^2(\theta) \) or \( \csc^2(\theta) = 1 + \cot^2(\theta) \)

Pythagorean Identities - Set of Three:

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

2. \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

3. \( 1 + \cot^2(\theta) = \csc^2(\theta) \)

These three Pythagorean Identities are all fundamentally derived from the original Pythagorean Theorem. They are all extremely useful in trigonometry. You should aim to memorize these and become fluent in using them.

Memorization Tip: Start by remembering \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Then, remember that dividing by \( \cos^2(\theta) \) leads to tangent and secant, and dividing by \( \sin^2(\theta) \) leads to cotangent and cosecant identities. This way, you can derive the second and third identities from the first one if you forget them.

5) Using Basic Trigonometric Identities - Examples and Verifications 🎯

Example 1: Simplifying using Identities

Simplify the expression: \( \sec(\theta) \cot(\theta) \sin(\theta) \).

Solution:

Use reciprocal and ratio identities to express everything in terms of sine and cosine.
\( \sec(\theta) = \frac{1}{\cos(\theta)}, \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
So, \( \sec(\theta) \cot(\theta) \sin(\theta) = \left(\frac{1}{\cos(\theta)}\right) \left(\frac{\cos(\theta)}{\sin(\theta)}\right) \sin(\theta) \)
Cancel out common factors: \( = \frac{1}{\cancel{\cos(\theta)}} \cdot \frac{\cancel{\cos(\theta)}}{\cancel{\sin(\theta)}} \cdot \cancel{\sin(\theta)} = 1 \)
Therefore, \( \sec(\theta) \cot(\theta) \sin(\theta) = 1 \).

Example 2: Verifying an Identity

Verify the identity: \( \frac{\sin^2(\theta)}{1 - \cos(\theta)} = 1 + \cos(\theta) \), for \( \cos(\theta) \neq 1 \).

Verification:

Start with the more complex side (left side) and try to simplify it to match the right side.
Left Side \( = \frac{\sin^2(\theta)}{1 - \cos(\theta)} \).
Use Pythagorean Identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \).
Substitute: \( = \frac{1 - \cos^2(\theta)}{1 - \cos(\theta)} \).
Factor the numerator: \( 1 - \cos^2(\theta) = (1 - \cos(\theta))(1 + \cos(\theta)) \) (difference of squares).
So, \( = \frac{(1 - \cos(\theta))(1 + \cos(\theta))}{1 - \cos(\theta)} \).
Cancel out common factor \( (1 - \cos(\theta)) \) (since \( \cos(\theta) \neq 1 \), it's not zero): \( = \frac{\cancel{(1 - \cos(\theta))}(1 + \cos(\theta))}{\cancel{1 - \cos(\theta)}} = 1 + \cos(\theta) \).
This is equal to the Right Side. Thus, the identity is verified.

Example 3: Using Pythagorean Identity to Simplify

Simplify the expression: \( \sec^2(\theta) - \tan^2(\theta) \).

Solution:

Recall the Pythagorean Identity: \( 1 + \tan^2(\theta) = \sec^2(\theta) \).
Rearrange this identity to get \( \sec^2(\theta) - \tan^2(\theta) \). Subtract \( \tan^2(\theta) \) from both sides.
\( \sec^2(\theta) - \tan^2(\theta) = 1 \).
Thus, \( \sec^2(\theta) - \tan^2(\theta) \) simplifies to just 1.

6) Practice Questions 🎯

6.1 Fundamental – Applying Basic Identities

1. State the three Reciprocal Identities for trigonometric functions.

2. State the two Ratio Identities for trigonometric functions.

3. State the three Pythagorean Identities. Which one is the primary identity from which the other two are derived?

4. Simplify: \( \csc(\theta) \sin(\theta) \).

5. Simplify: \( \frac{\tan(\theta)}{\sin(\theta)} \).

6. Simplify: \( \cos(\theta) \sec(\theta) + \sin(\theta) \csc(\theta) \).

7. Simplify: \( \csc^2(\theta) - \cot^2(\theta) \).

8. Simplify: \( (1 - \sin^2(\theta)) \).

9. Rewrite \( \tan(\theta) \) in terms of \( \sec(\theta) \) using a Pythagorean Identity (assume \( \theta \) is in the first quadrant so tangent is positive).

10. True or False: For any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = \sec^2(\theta) - \tan^2(\theta) \). Explain.

6.2 Challenging – Proofs & Complex Simplifications 💪🚀

1. Verify the identity: \( \frac{\cos(\theta)}{1 + \sin(\theta)} + \frac{\cos(\theta)}{1 - \sin(\theta)} = 2\sec(\theta) \).

2. Verify the identity: \( \tan(\theta) + \cot(\theta) = \sec(\theta) \csc(\theta) \).

3. Simplify the expression: \( \frac{\sec^2(\theta) - 1}{\sin^2(\theta)} \).

4. Simplify: \( \frac{\sin(\theta)}{1 + \cos(\theta)} + \frac{1 + \cos(\theta)}{\sin(\theta)} \).

5. (Conceptual) Explain, using geometric interpretations (e.g., unit circle or right triangles), why the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \) makes sense. Think about the relationship between tangent, secant, and the sides of a right triangle.

7) Summary 🎉

Trigonometric Identities: Equations true for all valid angles. Used for simplification, solving equations, and proofs.
Reciprocal Identities: Define \( \csc, \sec, \cot \) as reciprocals of \( \sin, \cos, \tan \).
Ratio Identities: Define \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
Pythagorean Identities: \( \sin^2(\theta) + \cos^2(\theta) = 1 \), \( 1 + \tan^2(\theta) = \sec^2(\theta) \), \( 1 + \cot^2(\theta) = \csc^2(\theta) \). Derived from Pythagorean Theorem.
Using Identities: Simplify expressions, verify other identities, solve equations. Key tool in trigonometry.

Excellent work! You've now learned the basic trigonometric identities – reciprocal, ratio, and Pythagorean identities! These are your fundamental tools for manipulating trigonometric expressions and solving trigonometric problems. Practice using these identities to simplify expressions and verify other relationships, and you'll become more and more comfortable with them! 🎭📐🚀