Trigonometric Functions Notes

Trigonometric Functions

• Study of relationships between lengths, angles of geometrical figures, and recurring cycle graphs.

• Advanced by mathematicians from various ancient civilizations.

• Modern trigonometry credited to Euler & Newton.

• Applications in meteorology, geology, physics, engineering, architecture, surveying, and radiology.

Trigonometry Review

• Find the value of a trigonometric ratio for any angle with and without technology.

• Find the exact values of $sin(\theta)$, $cos(\theta)$, and $tan(\theta)$

  • SOHCAHTOA (if sides are given).

  • Magic triangles for 30°, 45°, and 60°.

Right-Angled Triangles

• Hypotenuse: Longest side, opposite the 90° angle.

• Sine, cosine, and tangent rules are applicable only on right-angled triangles.

• Solving right-angled triangles requires trigonometric ratios and Pythagoras’ theorem.

Exact Values for Trigonometric Ratios of 30°, 45°, 60°

• Calculated using magic triangles and SOHCAHTOA.

• $sin 30° = \frac{1}{2}$, $cos 30° = \frac{\sqrt{3}}{2}$, $tan 30° = \frac{1}{\sqrt{3}}$

• $sin 60° = \frac{\sqrt{3}}{2}$, $cos 60° = \frac{1}{2}$, $tan 60° = \sqrt{3}$

• $sin 45° = \frac{1}{\sqrt{2}}$, $cos 45° = \frac{1}{\sqrt{2}}$, $tan 45° = 1$

Finding One Trigonometric Ratio Using Another

• Use Pythagoras’ theorem to find more sides and trig ratios if one sin/cos/tan ratio is known.

Radian Measure

• Alternative to degrees for measuring angles.

• More efficient for certain calculations.

• One radian is the angle made by wrapping one radius in an arc around a unit circle.

• $180° = \pi$ radians

• Degrees to radians: $x \times \frac{\pi}{180}$

• Radians to degrees: $x \times \frac{180}{\pi}$

Circle Calculations Using Radians

• Arc length ($l$) = $\theta r$ where $\theta$ is in radians.

• Sector area ($A$) = $\frac{1}{2} \theta r^2$

Unit Circle

• Center (0,0), radius 1 unit.

• Equation: $x^2 + y^2 = 1$

• Angles measured anticlockwise (positive) or clockwise (negative).

• $cos(\theta)$ is the x-coordinate, $sin(\theta)$ is the y-coordinate.

• $tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$

Exact Values and Symmetry Properties

• CAST rule indicates where trigonometric functions are positive.

• Tool #1: CAST

• Tool #2: Exact trigonometric values of $\frac{\pi}{6}$, $\frac{\pi}{4}$, and $\frac{\pi}{3}$.

• Tool #3: Draw using a bowtie.

Graphs of Trigonometric Functions

• Sine and cosine are periodic functions.

• Horizontal translation of $\frac{\pi}{2}$ transforms $y = cos(x)$ to $y = sin(x)$ and vice versa.

• Amplitude: Distance from center to max/min.

• Period (T): Horizontal distance between successive max/min points.

Finding Equations of Trig Graphs

• Determine a, k, and b from the graph.

• Identify sine or cosine, positive or negative.

• Write the whole equation.

Solving Trigonometric Equations

• Use DRAW (bowtie & base angle), magic triangles, and CAST.

• For difficult equations, look for a hidden quadratic.

• Pythagorean identity: $sin^2(A) + cos^2(A) = 1$

Modelling with Trig Equations

• Interpret a trig equation from a worded situation.

• Solve problems using trigonometric functions.