Chapter 7 Part 2: Trigonometric Identities

Welcome back to Professor Baker's Math Class! In this session, we continued our exploration of Chapter 7, focusing on proving trigonometric identities using a variety of techniques. Remember, the key to mastering these identities is practice, practice, practice! Let's review the topics we covered:

Topics Reviewed:

  • Proving Identities with Fundamental Identities: We tackled several problems using the core trigonometric identities. Here are some examples similar to those we did in class:
    • Problem Type 5: This involved manipulating expressions to match one side of the identity to the other. For instance, proving $\sec^2{x} + \csc^2{x} = \sec^2{x} \csc^2{x}$. The steps would involve rewriting in terms of $\sin$ and $\cos$ and combining fractions.
    • Problem Type 6: We may have looked at something like $\frac{\sec{x} + \cos{x}}{\sec{x} - \cos{x}} = \frac{1 + \cos^2{x}}{\sin^2{x}}$. This would involve multiplying the top and bottom by $\cos{x}$ to simplify the expression.
    • Problem Type 7: These problems often require creative algebraic manipulation. For example, to show $\frac{\tan{x}}{\sec{x}+1} = \frac{\sec{x}-1}{\tan{x}}$, we can multiply the numerator and denominator on the left side by $\sec{x}-1$.
  • Odd and Even Identities: A crucial tool for simplifying expressions! Remember these key identities:
    • $\sin(-u) = -\sin(u)$
    • $\cos(-u) = \cos(u)$
    • $\tan(-u) = -\tan(u)$
    • $\csc(-u) = -\csc(u)$
    • $\sec(-u) = \sec(u)$
    • $\cot(-u) = -\cot(u)$

    We used these to simplify expressions, such as proving $\sec(-x) - \sin(-x)\tan(-x) = \cos(x)$.

  • Sum and Difference Identities: These are essential for expanding trigonometric functions of sums or differences of angles:
    • $\sin(A + B) = \sin A \cos B + \cos A \sin B$
    • $\sin(A - B) = \sin A \cos B - \cos A \sin B$
    • $\cos(A + B) = \cos A \cos B - \sin A \sin B$
    • $\cos(A - B) = \cos A \cos B + \sin A \sin B$
    • $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
    • $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
  • Applying Sum and Difference Identities:
    • Problem Type 2 (Degrees): We worked on problems where we had to find the exact value of trigonometric functions for angles like $\cos(27^{\circ})\cos(18^{\circ}) - \sin(27^{\circ})\sin(18^{\circ})$. By recognizing this as $\cos(A+B)$, where $A = 27^{\circ}$ and $B = 18^{\circ}$, we could simplify it to $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$.
    • Problem Type 3: Similar to the above, but potentially involving $\tan$ or using the identities in reverse to simplify.
  • Proving Identities Using Sum and Difference Identities: This combined our knowledge! We manipulated expressions, expanded using sum/difference identities, and then simplified.

Remember, the most important thing is to understand the fundamental identities and how to manipulate them. Don't be afraid to experiment and try different approaches. Keep practicing, and you'll master these identities in no time!