Honors Precalculus 2nd Semester Comprehensive Semester Review

Right Triangle Trigonometry and Real-World Applications

  • Right Triangle Calculations:     * For a triangle where side b=30b = 30 and side c=34c = 34, trigonometric ratios (sine, cosine, and tangent) are used to determine missing side lengths and angle measures. All resulting values for sides and angles should be rounded to the nearest tenth.     * The Ladder Problem (Problem 3): A ladder with a length of 30 ft30\,ft is leaned against an office building. The base of this ladder is positioned precisely 17 ft17\,ft from the base of the building. To find the angle of elevation from the base of the ladder to the top, the cosine ratio is applied (cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}), and the final angle is rounded to the nearest tenth.     * The Watchtower Problem (Problem 4): An observer in a watchtower identifies a forest fire. The watchtower has a height of 25 m25\,m. The angle of depression from the tower to the fire is measured at 8∘8^{\circ}. To find the distance between the base of the tower and the fire, trigonometric ratios are employed, with the final distance rounded to the nearest tenth of a meter.

Oblique Triangles and the Law of Sines/Cosines

  • General Application: For triangles that do not contain a right angle, the Law of Sines or the Law of Cosines must be utilized to solve for missing side lengths and angle measures. Results should be rounded to the nearest tenth.
  • The Rotunda Problem (Problem 7): A shopping mall rotunda is designed in a triangular shape. A local scout troop intends to walk around the entire outer edge of this rotunda exactly 33 times to promote physical fitness. The total distance walked must be calculated to the nearest tenth of a foot using the Law of Sines or the Law of Cosines to determine the perimeter of the triangle before multiplying by the three laps.

The Unit Circle and Exact Values

  • Exact Value Determination: The review requires finding the exact values of all six trigonometric functions (sin⁡(θ)\sin(\theta), cos⁡(θ)\cos(\theta), tan⁡(θ)\tan(\theta), csc⁡(θ)\csc(\theta), sec⁡(θ)\sec(\theta), cot⁡(θ)\cot(\theta)) for specific angles on the unit circle:     * −7π6-\frac{7\pi}{6}     * π\pi     * 5π4\frac{5\pi}{4}
  • Identity Verification: A series of problems (11-16) focus on verifying trigonometric identities, requiring algebraic manipulation to show that both sides of an equation are equivalent.

Sum, Difference, and Double-Angle Identities

  • Sum and Difference Applications: These identities are used to find the exact values for angles that are not standard unit circle values:     * cos⁡(255∘)\cos(255^{\circ})     * sin⁡(105∘)\sin(105^{\circ})
  • Quadrant-Specific Trigonometry: If sin⁡(α)=−35\sin(\alpha) = -\frac{3}{5} and the angle α\alpha is located in Quadrant IV (where cosine is positive and sine is negative), the following must be derived:     * The exact value of cos⁡(2α)\cos(2\alpha) using double-angle formulas.     * The exact value of tan⁡(2α)\tan(2\alpha) using double-angle formulas.

Solving Trigonometric Equations

  • Degree Interval Solution ([0,360∘)[0, 360^{\circ})): Equations must be solved with the calculator in degree mode, rounding to the nearest tenth of a degree.     * 13csc⁡(x)+25=5csc⁡(x)+713\csc(x) + 25 = 5\csc(x) + 7     * 24cos⁡2(x)+11cos⁡(x)+1=024\cos^2(x) + 11\cos(x) + 1 = 0 (Quadratic form)     * 12cot⁡(2x)−7=2cot⁡(2x)−1012\cot(2x) - 7 = 2\cot(2x) - 10
  • Radian Interval Solution ([0,2π)[0, 2\pi)): Find exact solutions for the following equations:     * 5cot⁡(x)+23=−2cot⁡(x)−535\cot(x) + 2\sqrt{3} = -2\cot(x) - 5\sqrt{3}     * 4sin⁡2(x)−8cos⁡(x)+1=04\sin^2(x) - 8\cos(x) + 1 = 0     * sin⁡(2x)+cos⁡(x)=0\sin(2x) + \cos(x) = 0     * 3sec⁡2(x)=43\sec^2(x) = 4

Graphing Trigonometric Functions

  • Function Analysis: For each function, the amplitude, period, and vertical shift must be identified before graphing two full cycles.
  • Function 1: f(x)=−2sin⁡(x4)+2f(x) = -2\sin(\frac{x}{4}) + 2     * Amplitude: Reflects the vertical stretch and is determined by the absolute value of the leading coefficient.     * Period: Calculated as 2π∣b∣\frac{2\pi}{|b|}, which for this function is 2π1/4=8π\frac{2\pi}{1/4} = 8\pi.     * Vertical Shift: Indicated by the constant added to the function (+2+2).
  • Function 2: f(x)=3cos⁡(2x)−3f(x) = 3\cos(2x) - 3     * Amplitude: 33.     * Period: 2π2=π\frac{2\pi}{2} = \pi.     * Vertical Shift: −3-3.

Comprehensive Trigonometric Formula Reference

  • Law of Sines: sin⁡(A)a=sin⁡(B)b=sin⁡(C)c\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
  • Law of Cosines:     * a2=b2+c2−2bc⋅cos⁡(A)a^2 = b^2 + c^2 - 2bc \cdot \cos(A)     * b2=a2+c2−2ac⋅cos⁡(B)b^2 = a^2 + c^2 - 2ac \cdot \cos(B)     * c2=a2+b2−2ab⋅cos⁡(C)c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
  • Reciprocal Identities:     * sin⁡(θ)=1csc⁡(θ)\sin(\theta) = \frac{1}{\csc(\theta)} and csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}     * cos⁡(θ)=1sec⁡(θ)\cos(\theta) = \frac{1}{\sec(\theta)} and sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}     * tan⁡(θ)=1cot⁡(θ)\tan(\theta) = \frac{1}{\cot(\theta)} and cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}
  • Quotient Identities:     * tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}     * cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
  • Pythagorean Identities:     * sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1     * tan⁡2(θ)+1=sec⁡2(θ)\tan^2(\theta) + 1 = \sec^2(\theta)     * cot⁡2(θ)+1=csc⁡2(θ)\cot^2(\theta) + 1 = \csc^2(\theta)
  • Cofunction Identities:     * sin⁡(θ)=cos⁡(π2−θ)\sin(\theta) = \cos(\frac{\pi}{2} - \theta)     * cos⁡(θ)=sin⁡(π2−θ)\cos(\theta) = \sin(\frac{\pi}{2} - \theta)     * tan⁡(θ)=cot⁡(π2−θ)\tan(\theta) = \cot(\frac{\pi}{2} - \theta)     * csc⁡(θ)=sec⁡(π2−θ)\csc(\theta) = \sec(\frac{\pi}{2} - \theta)     * sec⁡(θ)=csc⁡(π2−θ)\sec(\theta) = \csc(\frac{\pi}{2} - \theta)     * cot⁡(θ)=tan⁡(π2−θ)\cot(\theta) = \tan(\frac{\pi}{2} - \theta)
  • Even-Odd Identities:     * sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta); csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta)     * cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta); sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta)     * tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta); cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta)
  • Sum and Difference Identities:     * sin⁡(A±B)=sin⁡(A)cos⁡(B)±cos⁡(A)sin⁡(B)\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)     * cos⁡(A±B)=cos⁡(A)cos⁡(B)∓sin⁡(A)sin⁡(B)\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)     * tan⁡(A±B)=tan⁡(A)±tan⁡(B)1∓tan⁡(A)tan⁡(B)\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}
  • Double-Angle Identities:     * sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\sin(2\theta) = 2\sin(\theta)\cos(\theta)     * cos⁡(2θ)=cos⁡2(θ)−sin⁡2(θ)=1−2sin⁡2(θ)=2cos⁡2(θ)−1\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1     * tan⁡(2θ)=2tan⁡(θ)1−tan⁡2(θ)\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}
  • Half-Angle Identities:     * sin⁡(θ2)=±1−cos⁡(θ)2\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}     * cos⁡(θ2)=±1+cos⁡(θ)2\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}     * tan⁡(θ2)=±1−cos⁡(θ)1+cos⁡(θ)=1−cos⁡(θ)sin⁡(θ)=sin⁡(θ)1+cos⁡(θ)\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}