Welcome to More about Trigonometry!

In your junior years, you learned how to find sides and angles in right-angled triangles. But what if the triangle isn't right-angled? Or what if you are working in 3D space? This chapter is your toolkit for solving these puzzles. Trigonometry is the "math of waves and rotations" used in everything from building bridges to designing video games.

Don't worry if this seems tricky at first! We will break it down into four simple parts: the behavior of trig functions, solving equations, new triangle formulas, and exploring the 3D world.


1. Trig Functions and Their Graphs

In senior math, we stop looking at sine, cosine, and tangent as just ratios and start looking at them as functions that repeat forever.

The Unit Circle and the CAST Rule

Imagine a circle with a radius of 1. As you move around it, your coordinates represent \(\cos \theta\) and \(\sin \theta\). To remember which function is positive in which quadrant, use the CAST rule (starting from the bottom-right quadrant and moving counter-clockwise):

  • Quadrant I (0° to 90°): All are positive.
  • Quadrant II (90° to 180°): Sine only is positive.
  • Quadrant III (180° to 270°): Tangent only is positive.
  • Quadrant IV (270° to 360°): Cosine only is positive.

Mnemonic: All Students Take Chemistry!

Graphs and Periodicity

Trig functions are periodic, meaning they repeat their pattern every few degrees.

  • Sine (\(\sin \theta\)) and Cosine (\(\cos \theta\)): Repeat every 360°. Their values always stay between -1 and 1.
  • Tangent (\(\tan \theta\)): Repeats every 180°. It can go up to infinity or down to negative infinity!

Quick Review Box:
The Maximum value of \(\sin \theta\) and \(\cos \theta\) is 1.
The Minimum value of \(\sin \theta\) and \(\cos \theta\) is -1.

Key Takeaway: Trig functions are circular patterns. Knowing the quadrant (CAST rule) tells you if the answer is positive or negative.


2. Solving Trigonometric Equations

Solving these is just like solving for \(x\), but you have to find all possible angles between 0° and 360°.

Step-by-Step for Simple Equations:

Example: Solve \(2 \sin \theta = 1\)
  1. Isolate the function: \(\sin \theta = 0.5\).
  2. Find the Reference Angle (\(\alpha\)): Use your calculator \(\sin^{-1}(0.5) = 30°\).
  3. Check the Sign: \(\sin \theta\) is positive. According to CAST, this happens in Quadrant I and II.
  4. Calculate the Final Angles:
    Q1: \(\theta = 30°\)
    Q2: \(\theta = 180° - 30° = 150°\)

Equations Transformable to Quadratics

Sometimes you might see \(2 \sin^2 \theta + 5 \sin \theta + 2 = 0\).
Pro-tip: Treat \(\sin \theta\) like \(y\). Solve the quadratic \(2y^2 + 5y + 2 = 0\) first, then find the angles for \(y\).

Did you know? If you get an answer like \(\sin \theta = 2\), there is no solution! Remember, sine and cosine can never be larger than 1.

Key Takeaway: Always check if your angles fall within the requested range (usually 0° to 360°).


3. Advanced Formulas for Triangles

These formulas work for any triangle, not just right-angled ones!

The Sine Formula

Use this when you have "opposite pairs" of sides and angles:
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

The Cosine Formula

Use this when you have three sides (SSS) or two sides and the angle between them (SAS):
\(a^2 = b^2 + c^2 - 2bc \cos A\)

Finding Area

  • SAS Method: Area = \(\frac{1}{2} ab \sin C\)
  • Heron’s Formula (SSS Method): Use this if you only know the three sides \(a\), \(b\), and \(c\).
    First, find the semi-perimeter: \(s = \frac{a+b+c}{2}\)
    Then: Area = \(\sqrt{s(s-a)(s-b)(s-c)}\)

Key Takeaway: Use the Sine Formula for pairs; use the Cosine Formula for "side-angle-side" layouts.


4. Trigonometry in 3D Space

This is often the most challenging part, but it’s just about finding the right 2D triangle hidden inside a 3D shape.

Key Terms to Master:

  • Projection: Imagine a light shining directly from above. The "shadow" a line casts on a floor is its projection.
  • Angle between a line and a plane: It is the angle between the line and its shadow (projection) on that plane.
  • Angle between two planes: Find two lines (one in each plane) that are both perpendicular to the intersection line. The angle between these two lines is the angle between the planes.

The Theorem of Three Perpendiculars

This sounds scary, but think of it as a 3D "corner." If a line is perpendicular to a plane, and a second line on that plane is perpendicular to a third line, then the line connecting them is also perpendicular. It helps us find right-angled triangles in 3D diagrams!

Common Mistake to Avoid: Don't just pick any two lines to find the angle between planes. They must meet at the same point on the intersection line and both be at 90° to it.

Key Takeaway: When solving 3D problems, draw the 2D triangles out separately. It makes the math much easier to see!


Summary Checklist

Can you:
1. Use the CAST rule to find the sign of a trig ratio?
2. Identify the period and max/min values of a graph?
3. Solve trig equations for multiple angles between 0° and 360°?
4. Choose between Sine and Cosine formulas for non-right triangles?
5. Identify the "angle of inclination" in a 3D pyramid or prism?

Keep practicing! Trigonometry is a skill that gets much easier once you recognize the patterns. You've got this!