Trigonometry and Pre-Calculus Problem Solving Techniques
Trigonometry and Triangle Calculations
1) Find Tangent Theta: Given sine theta and its quadrant, write sine as a fraction (opposite/hypotenuse). Find the adjacent side using the Pythagorean Theorem, adjust the sign based on the quadrant, and calculate tangent as opposite/adjacent.
2) Solve Triangle (Law of Sines): Use the formula: \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.
3) Solve Triangle: Apply the same Law of Sines method as described above.
15) SOH CAH TOA: Use basic trigonometric ratios for right-angled triangles.
22) Find the Period: Identify the number in front of x (this is b). For sine or cosine, the period is 2π ÷ b. For tangent, the period is π ÷ b.
Solving Equations and Parabola Width
4) Solve Equation for a Given Interval: Solve the trigonometric equation and find the reference angle. Use quadrant rules to obtain all solutions. Add or subtract 360° if needed, keeping only the answers within the given interval.
5) Solve Equation for a Given Interval (Radians): Follow the same process, but ensure the graph is in RADIAN mode.
6) Find the Width of a Parabola: Use the formula (x-h)^2 = 4p(y-k). Rewrite the equation in standard form and find 4p, which represents the width.
Resultant Force and Vector Magnitudes
7) Magnitude of Resultant Force: Given two forces and the angle they form, use the formula: R = \sqrt{A^2 + B^2 + 2AB\cos\theta}. Plug in both forces (A, B) and the angle theta. Square, multiply, add, and calculate the cosine; finally, take the square root.
18) Vector Magnitude: Calculate the magnitude using the formula |\vec{v}| = \sqrt{a^2 + b^2}.
Real-World Math Applications and Bearings
8) Bearing of a Ship: A ship sails 75km. Answer: The distance from point A is 92km.
9) Problem 9: This problem is self-explanatory.
13) Tunnel and Truck Question: Answer: The height of the tunnel at 10 ft from the center is 18.2 ft. Yes, the truck clears the tunnel with 4 ft of space.
14) Planes Flying Question: Answer: After 4 hours, the distance is 2480 miles with no radio contact.
Functions and Conic Sections
10) Function Evaluation: Plug the value 3 into the function f.
11) Maximum Value: Solve the word problem to find the maximum value.
12) Unit Conversions: Perform conversions similar to those used in chemistry and physics.
16 & 17) Guessing: Content to be determined.
25) Conic Section Vertices:
- Put the equation in standard form.
- Find the center (h, k).
- Find a² and take the square root to get a.
- Calculate the vertices:
- Horizontal: (h ± a, k)
- Vertical: (h, k ± a)