Trigonometric Exact Values

Introduction

Purpose: Solving trigonometric equations requires the use of exact values and reference angles to analyse periodic phenomena. Reference angles and periodicity are essential for identifying solutions in both theoretical and practical contexts, such as sound waves or alternating currents.

Exact Trigonometric Values for Common Angles

Grasping the exact trigonometric values for angles like 0°, 30°, 45°, 60°, and 90° is fundamental. These angles form the basis for trigonometric calculations through special triangles.

Table of Exact Values

Angle (°)	Sine	Cosine	Tangent
0	0	1	0
30	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90	1	0	undefined

Importance of these Angles

• Streamlines calculations in trigonometry.
• Employed in designing structures and animations, such as aligning edges and timing movements in 3D models.

Introduction to the Unit Circle

The unit circle: a circle with a radius of 1, centred at the origin, is crucial for calculating exact trigonometric values.

Understanding Quadrants

The unit circle is divided into four quadrants:

• Quadrant I: All functions positive.
• Quadrant II: Sine positive.
• Quadrant III: Tangent positive.
• Quadrant IV: Cosine positive.

Derivation from Special Triangles

• 30°-60°-90° Triangle: Fundamental to determining exact trigonometric values.

  • Side Ratios: 1 : $\sqrt{3}$ : 2

• 45°-45°-90° Triangle: Integral to the unit circle framework.

  • Side Ratios: 1 : 1 : $\sqrt{2}$

Mnemonic Devices and Techniques

• SOHCAHTOA: A mnemonic to aid in recalling function definitions.

  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent

• Active Learning Challenge: Calculate the sine of 30° without reference.

  • Hint: Recall it is half of a triangle's hypotenuse.

Conceptual Understanding vs Memorisation

• Emphasise geometric derivations for deeper insights.
• Interactive Tools: Utilise software like GeoGebra for visual learning.

Compound Angles: Understanding

Compound angles simplify complex problems by breaking down larger angles into smaller, known angles.

Essential Compound Angle Formulas

• sin(A ± B):

  • Formula: $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$

• cos(A ± B):

  • Formula: $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$

• tan(A ± B):

  • Formula: $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$

Example Problems

Example 1: Calculate $\sin(75°)$ using $\sin(45° + 30°)$.

1. Decompose the Angle: Break down $75°$ as $45° + 30°$.
2. Apply the Formula:
   • $\sin(75°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°)$
3. Substitute Values:
   • $= \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)$
4. Simplify:
   • $= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Related Angles and Reference Angles

• Reference Angle: The smallest angle between the terminal side and the x-axis.

• Related Angle: Angles with terminal sides across different rotations, such as multiples of $360^\circ$. These help reduce complex angles to simpler components.

Quadrant Rule

• Determining Signs: The signs of functions depend on the quadrant of the angle.

  chatImportant

  • 1st Quadrant: All positive
  • 2nd Quadrant: Sine positive, others negative
  • 3rd Quadrant: Tangent positive, others negative
  • 4th Quadrant: Cosine positive, others negative

Conclusion on Solving Trigonometric Equations

Example: Solving $\sin x = \frac{1}{2}$

• Step 1: Identify the reference angle:
  • $\theta = \frac{\pi}{6}$
• Step 2: Solutions in $[0, 2\pi]$:
  • $x = \frac{\pi}{6}, \frac{5\pi}{6}$
• Step 3: Extend solutions using periodicity:
  • $x = \frac{\pi}{6} + 2k\pi, x = \frac{5\pi}{6} + 2k\pi$

Common Pitfalls and Misunderstandings

• Mistaking angles and improper application of formulas.
• Use mnemonics like All Students Take Calculus to remember which functions are positive in which quadrant.

Master these concepts to build competence in advanced trigonometry.