Comprehensive Trigonometry Study Guide

Solving Triangles and Trigonometric Foundations

• To solve a triangle means to determine all unknown side lengths and angle measures.

• Pythagorean Theorem Application: In right-angled triangles, the relationship between sides is given by $a^2 + b^2 = c^2$.     - Example: Given a hypotenuse of $15\,cm$ and one side of $12\,cm$, the third side $x$ is calculated as $x = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9\,cm$.

• Primary Trigonometric Ratios (Right Triangles):   

•  - $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$     - $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$     - $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Trigonometry of Acute and Non-Right Triangles

• These laws apply to triangles that do not necessarily contain a $90^{\circ}$ angle.

• Sine Law:     - Formula: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Usage Prerequisites: Requires two angles and one side (AAS or ASA) or two sides and an opposite angle (SSA).    

• Example: In $\triangle XYZ$, if $\angle Y = 84^{\circ}$ and $\angle Z = 23^{\circ}$ with side $XY = 4\,cm$:         - Find $\angle X = 180^{\circ} - 84^{\circ} - 23^{\circ} = 73^{\circ}$.         - Find side $XZ$: \frac{XZ}{\sin(84^{\circ})} = \t\frac{4}{\sin(27^{\circ})} (Note: transcript indicates a logic search for $\angle Z$, results in $XZ = \frac{4 \times \sin(69^{\circ})}{\sin(27^{\circ})} \approx 8.23\,cm$).

• Cosine Law:     - Formula: $a^2 = b^2 + c^2 - 2bc \cos(A)$.     - Rearranged for Angle: $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$.     

• Usage Prerequisites: Requires two sides and a contained angle (SAS) to find the third side, or three sides (SSS) to find an angle.     

• Example (SSS): Given sides $a=19$, $b=14$, and $c=17$, find $\angle B$:         - $19^2 = 14^2 + 17^2 - 2(14)(17) \cos(B)$         - $\cos(B) = \frac{14^2 + 17^2 - 19^2}{2(14)(17)} = \frac{196 + 289 - 361}{476} = \frac{124}{476}$         - $\angle B = \cos^{-1}(\frac{124}{476}) \approx 74.9^{\circ}$.

Describing Direction and Bearings

• Angle of Elevation: Measured upward from the horizontal line.

• Angle of Depression: Measured downward from the horizontal line.

• True Bearing:     - Measured clockwise from North ($000^{\circ}$).     - Always expressed as 3 digits (e.g., $020^{\circ}$ for $20^{\circ}$ East of North).

• Quadrant Bearing:     - Measured East or West from the North-South line.     - Always an acute angle.     - Format Example: $S30^{\circ}E$ (Start South, turn $30^{\circ}$ toward East) or $N45^{\circ}W$.

Angles in the Cartesian Plane

• Standard Position: An angle where the vertex is at the origin $(0,0)$ and the initial arm lies on the positive $x$-axis ($0^{\circ}$).

• Rotation Direction:     - Positive Angle: Counter-clockwise rotation.     - Negative Angle: Clockwise rotation.

• Quadrants:     - Quadrant I: $0^{\circ}$ to $90^{\circ}$     - Quadrant II: $90^{\circ}$ to $180^{\circ}$     - Quadrant III: $180^{\circ}$ to $270^{\circ}$     - Quadrant IV: $270^{\circ}$ to $360^{\circ}$

• Related Acute Angle (RAA) or $\beta$:     - The acute angle (between $0^{\circ}$ and $90^{\circ}$) formed between the terminal arm and the $x$-axis.     - If $\theta = 150^{\circ}$ (Quadrant II), $\beta = 180^{\circ} - 150^{\circ} = 30^{\circ}$.

• Co-terminal Angles:     - Angles that share the same terminal arm location.     - Found by adding or subtracting multiples of $360^{\circ}$.     - Example: Co-terminal angles for $150^{\circ}$ include $510^{\circ}$ ($150 + 360$) and $-210^{\circ}$ ($150 - 360$).

• Principal Angle: The specific co-terminal angle $\theta$ such that 0^{\circ} \le \theta < 360^{\circ}.

The CAST Rule and General Trig Ratios

• The CAST rule identifies which primary trig ratios are positive in each quadrant:     - C (Quadrant IV): Cosine is positive.     - A (Quadrant I): All (Sine, Cosine, Tangent) are positive.     - S (Quadrant II): Sine is positive.     - T (Quadrant III): Tangent is positive.

• General Point Coordinates: For a point $P(x, y)$ on the terminal arm with distance $r$ from the origin ($r = \sqrt{x^2 + y^2}$):     - $\sin(\theta) = \frac{y}{r}$     - $\cos(\theta) = \frac{x}{r}$     - $\tan(\theta) = \frac{y}{x}$

• Example calculation with point $P(-3, 4)$:     - $r = \sqrt{(-3)^2 + 4^2} = 5$.    

 - Find RAA ($\beta$): $\tan(\beta) = |\frac{4}{-3}| \Rightarrow \beta = \tan^{-1}(\frac{4}{3}) \approx 53^{\circ}$.    

 - Since the point is in QII, the principal angle $\theta = 180^{\circ} - 53^{\circ} = 127^{\circ}$.

Special Triangles and the Unit Circle

• Special Triangles (Method #1): Used to find exact values.     - $45^{\circ} - 45^{\circ} - 90^{\circ}$:

• Sides are $1, 1, \sqrt{2}$.         - $\sin(45^{\circ}) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$         - $\cos(45^{\circ}) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$       

•  - $\tan(45^{\circ}) = 1$     

- $30^{\circ} - 60^{\circ} - 90^{\circ}$: Sides are $1$ (opposite $30^{\circ}$), $\sqrt{3}$ (opposite $60^{\circ}$), and $2$ (hypotenuse).         

- $\sin(30^{\circ}) = \frac{1}{2}$; $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$; $\tan(30^{\circ}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$       

 - $\sin(60^{\circ}) = \frac{\sqrt{3}}{2}$; $\cos(60^{\circ}) = \frac{1}{2}$; $\tan(60^{\circ}) = \sqrt{3}$

• Unit Circle (Method #2): A circle with radius $r = 1$. Coordinates are $(\cos(\theta), \sin(\theta))$.     - Quadrantal Angles:      

  $0^{\circ} / 360^{\circ}: (1, 0) \Rightarrow \sin=0, \cos=1, \tan=0$         - $90^{\circ}: (0, 1) \Rightarrow \sin=1, \cos=0, \tan=\text{Undefined}$         - $180^{\circ}: (-1, 0) \Rightarrow \sin=0, \cos=-1, \tan=0$         - $270^{\circ}: (0, -1) \Rightarrow \sin=-1, \cos=0, \tan=\text{Undefined}$

The Ambiguous Case of the Sine Law (SSA)

• Occurs when given two sides and a non-included angle (SSA), specifically if the given angle $A$ is acute and the side opposite ($a$) is shorter than the adjacent side ($b$).

• Possibilities for Acute $A$:    

 - If a < b \sin(A): No triangle exists (side $a$ is too short to reach the base).     

- If $a = b \sin(A)$: One right-angled triangle exists.     

- If b \sin(A) < a < b: Two possible triangles exist (one acute, one obtuse).     

- If $a \ge b$: Only one triangle exists.

• Example Case (2 Solutions): Given $a=15$, $b=23$, $\angle A = 32^{\circ}$.     

- Calculate height $h = 23 \sin(32^{\circ}) \approx 12.19$.    

 - Since 12.19 < 15 < 23, there are two solutions.     - $\sin(B) = \frac{23 \sin(32^{\circ})}{15} \Rightarrow B_1 \approx 54.35^{\circ}$.     - The second possible angle is the supplement: $B_2 = 180^{\circ} - 54.35^{\circ} = 125.65^{\circ}$.

Solving Trigonometric Equations

• Example 1: Find $\theta$ for $0^{\circ} \le \theta \le 360^{\circ}$ given $\sin(\theta) = 0.75$.     - $\text{RAA} = \sin^{-1}(0.75) \approx 48.6^{\circ}$.     - Sine is positive in Q1 and Q2.     

• $\theta_1 = 48.6^{\circ}$.      $\theta_2 = 180^{\circ} - 48.6^{\circ} = 131.4^{\circ}$.

• Example 2: Find $\theta$ for $0^{\circ} \le \theta \le 540^{\circ}$ given $\cos(\theta) = -0.5$.     - $\text{RAA} = \cos^{-1}(0.5) = 60^{\circ}$.     - Cosine is negative in Q2 and Q3.     - Q2: $180^{\circ} - 60^{\circ} = 120^{\circ}$.     - Q3: $180^{\circ} + 60^{\circ} = 240^{\circ}$.     - Extension to $540^{\circ}$: Check co-terminals ($120^{\circ} + 360^{\circ} = 480^{\circ}$. $240^{\circ} + 360^{\circ} = 600^{\circ}$, which is out of range).     - Solutions: $120^{\circ}, 240^{\circ}, 480^{\circ}$.

• Equivalent Trig Values:     - $\cos(\theta) = \cos(250^{\circ})$: Since $250^{\circ}$ is in Q3 ($RAA=70^{\circ}$), and Cosine is negative in Q2 and Q3, the other solution is in Q2: $\theta = 180^{\circ} - 70^{\circ} = 110^{\circ}$.     - $\tan(130^{\circ}) = \tan(\theta)$: Since $130^{\circ}$ is in Q2 ($RAA=50^{\circ}$), and Tangent is negative in Q2 and Q4, the other solution is in Q4: $\theta = 360^{\circ} - 50^{\circ} = 310^{\circ}$.