Trigonometric Identities and Equations

Double-Angle Formulas: - $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ - $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1$ - $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$
Reduction Formulas (Power-Reducing): - $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ - $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$ - $\tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}$
Half-Angle Formulas: - $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$ - $\text{cos}(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$ - $\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\text{sin}(\theta)}{1 + \text{cos}(\theta)} = \frac{1 - \text{cos}(\theta)}{\text{sin}(\theta)}$ - The $±$ sign depends on the quadrant where $\frac{\theta}{2}$ terminates.
Application: Bicycle Ramps: - Competition ramps vary based on skill. - Advanced ramp angle $\theta$ : $\tan(\theta) = \frac{5}{3}$ . - Novice ramp angle measures half of that ( $\frac{\theta}{2}$ ). - Calculated novice angle: Approximately $29.5^{∘}$ .

Historical Application (Pyramid Height): - Thales of Miletus (c. 625-547 BC) computed the height of the Great Pyramid of Giza. - Method: Theory of similar triangles using the shadow of his staff. - Logic: When the staff's shadow equaled its height, the pyramid's shadow equaled its actual height.
Solving Strategies: - Find specific values over an interval or all possible solutions (using $+ 2\text{π}k$ for periodic functions). - Period considerations: Sine and Cosine ( $2\text{π}$ ), Tangent ( $\text{π}$ ).
Algebraic Techniques: - Linear equations (isolate the function). - Quadratic forms ( $ax^2 + bx + c = 0$ ): Use factoring or the Quadratic Formula ( $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ). - Note: Sine and Cosine values must be within $[-1, 1]$ ; values outside this range are invalid solutions.
Multiple Angle Equations: - If solving for $\sin(n\theta)$ , one must go around the unit circle $n$ times to find all solutions on the internal $[0, 2\text{π})$ .
Real-World Applications: - London Eye Ferris Wheel: Replacement of a cable. Anchor height of $69.5\,m$ , distance of $23\,m$ from base. Calculated cable length: $73.2\,m$ ; Angle of elevation: $71.7^{∘}$ . - OSHA Standards: Ladder placement requiring $1\,ft$ from wall for every $4\,ft$ of ladder length. Resulting angle with ground is always $\approx 75.5^{∘}$ .

General Sherman Tree: World's largest living tree by volume ( $274.9\,ft$ tall); researchers use angle of elevation for measurement.
Oblique Triangles: Defined as triangles that are not right triangles. Solving requires at least three values, including one side.
Problem Situations: - ASA (Angle-Side-Angle). - AAS (Angle-Angle-Side). - SSA (Side-Side-Angle) - The Ambiguous Case.
Law of Sines Formula: - $\frac{a}{\sin(\text{α})} = \frac{b}{\sin(\text{β})} = \frac{c}{\sin(\text{γ})}$
The Ambiguous Case (SSA): - Can result in no triangle, one triangle, or two distinct triangles.
Area of Oblique Triangle: - $\text{Area} = \frac{1}{2}bc\sin(\text{α}) = \frac{1}{2}ac\sin(\text{β}) = \frac{1}{2}ab\sin(\text{γ})$
Application: Air Traffic Control: - Altitude tracking between two radar stations $20\,miles$ apart. Angles of elevation are $35^{∘}$ and $15^{∘}$ . Calculated altitude: $3.9\,miles$ .

$22^{∘}$ ; Initial height: $3\,ft$ . - After 2 seconds: Ball is $198\,ft$ away and $137\,ft$ high. - Ground impact: At $t \approx 3.32\,seconds$ . - Distance wall result: At $400\,ft$ (deep park), the ball is $141.8\,ft$ high, clearing a $10\,ft$ wall easily for a home run.