04.24.2019: One Lesson of Math - Trigonometric Equations and Identities, 3/6: Trigonometric Identit

Today's soundtrack is Dream Theater: Falling Into Infinity, probably their most radio-friendly album.

You may have noticed that this series is titled "Trigonometric Equations and Identities." So far, we've been working with equations; today, we'll be learning about trigonometric identities. Let's start by differentiating the two.

An equation is only true for certain values of its variables.

An identity is true for any values of its variables.

The following are the identities that we should be familiar with:

  • Reciprocal Identities

  1. secθ = 1 / cosθ

  2. cscθ = 1 / sinθ

  3. cotθ = 1 / tanθ

  • Quotient Identities

  1. tanθ = sinθ / cosθ

  2. cotθ = cosθ / sinθ

  • Pythagorean Identities

  1. sin²θ + cos²θ = 1

  2. 1 + cot²θ = csc²θ

  3. 1 + tan²θ = sec²θ

All of these identities will be true for all values of θ.

We can prove identities by breaking down complex terms into simple terms through simplifying. We start with a two-sided equation. We split the equation with a bar down the middle, then simplify the more complicated side. After that, we simplify the easier side, and show that the two values are equal.

While we prove identities, we will be asked to simplify rational expressions using this method:

  • Identify the denominator of the top term

  • Identify the denominator of the bottom term

  • Combine them as a common denominator

  • Multiply the top term by the common denominator

  • Multiply the bottom term by the common denominator

  • Simplify

If we are asked to verify an equation for a value, we simply substitute the value in for the variable and solve by breaking down the identities as discussed above. We can also solve graphically by using technology and checking for intersects. If the graphs overlap at all points, then the value is verified.

That's it for today; next time, we'll tackle part 2 of this lesson.