Professor Baker's Math Class - Chapter 7 Part 1 (10-24-2023)

Welcome to Chapter 7! Today's lesson is packed with fundamental trigonometric identities and techniques that will empower you to simplify expressions, verify identities, and confidently prove trigonometric relationships. Let's dive in!

Topics Covered:

  • Using Cofunction Identities: These identities allow you to express trigonometric functions in terms of their complements. Remember, the cofunction of sine is cosine, tangent is cotangent, and secant is cosecant. For example: $sin(x) = cos(\frac{\pi}{2} - x)$
  • Finding Values of Trigonometric Functions Given Information About an Angle (Problem Type 3): This involves using given information (like the quadrant and the value of one trigonometric function) to find the values of other trigonometric functions. Key is to visualize the angle in the correct quadrant and use the Pythagorean theorem and definitions of trigonometric functions.
  • Simplifying Trigonometric Expressions with Reciprocal and Quotient Identities: Reciprocal identities: $sin(u) = \frac{1}{csc(u)}$, $cos(u) = \frac{1}{sec(u)}$, $tan(u) = \frac{1}{cot(u)}$. Quotient Identities: $tan(u) = \frac{sin(u)}{cos(u)}$, $cot(u) = \frac{cos(u)}{sin(u)}$. We can use these to rewrite and simplify complex expressions. For example, simplifying $\frac{csc(x)}{sec(x)}$ can be done as follows: $$ \frac{csc(x)}{sec(x)} = \frac{\frac{1}{sin(x)}}{\frac{1}{cos(x)}} = \frac{1}{sin(x)} \cdot \frac{cos(x)}{1} = \frac{cos(x)}{sin(x)} = cot(x) $$
  • Simplifying Trigonometric Expressions with Pythagorean Identities: Master the core Pythagorean identity: $sin^2(u) + cos^2(u) = 1$. From this, you can derive $tan^2(u) + 1 = sec^2(u)$ and $cot^2(u) + 1 = csc^2(u)$. These are powerful tools for simplification. For instance, simplifying $(1 + cot^2(x))cos^2(x)$: $$(1 + cot^2(x))cos^2(x) = (csc^2(x))cos^2(x) = \frac{1}{sin^2(x)} \cdot cos^2(x) = \frac{cos^2(x)}{sin^2(x)} = cot^2(x)$$
  • Verifying Trigonometric Identities (Problem Types 1, 2, and 3): Verifying means showing that one side of an equation is equal to the other. Use algebraic manipulation and trigonometric identities to transform one side until it matches the other.
  • Proving an Identity Using Fundamental Trigonometric Identities (Problem Types 1, 2, and 3): Similar to verifying, but often requires more strategic application of identities. Remember to choose the more complex side to simplify. Common strategies involve converting everything to sines and cosines, factoring, or using conjugate multiplication.
  • Proving Trigonometric Identities Using Odd and Even Identities: These identities describe how trigonometric functions behave with negative angles. Key identities: $sin(-u) = -sin(u)$, $cos(-u) = cos(u)$, $tan(-u) = -tan(u)$. $csc(-u)=-csc(u)$, $sec(-u) = sec(u)$, $cot(-u)=-cot(u)$. Recognizing even and odd properties can lead to elegant simplifications.
  • Sum and Difference Identities: These identities are useful when dealing with trigonometric functions of sums or differences of angles, $sin(u+v) = sin(u)cos(v)+cos(u)sin(v)$, $cos(u+v) = cos(u)cos(v)-sin(u)sin(v)$, $tan(u+v) = \frac{tan(u) + tan(v)}{1-tan(u)tan(v)}$
  • Double Angle Identities: $sin(2u) = 2sin(u)cos(u)$, $cos(2u) = cos^2(u) - sin^2(u)$, $tan(2u) = \frac{2tan(u)}{1-tan^2(u)}$

Keep practicing, and you'll master these trigonometric techniques in no time! Good luck!