Problem 1-5: Understanding Trigonometric Ratios
These problems focus on the fundamental concept of trigonometric ratios, including sine, cosine, and tangent. The goal is to understand how these ratios relate to the sides of a right triangle. To solve these problems, you need to recall the definitions of sine, cosine, and tangent, as well as the Pythagorean identity.- Recall the definitions of sine, cosine, and tangent: sin(A) = opposite side / hypotenuse, cos(A) = adjacent side / hypotenuse, tan(A) = opposite side / adjacent side.
- Use the Pythagorean identity: sin^2(A) + cos^2(A) = 1.
- Apply trigonometric ratios to solve for unknown side lengths.
cos(A) = adjacent side / hypotenuse = 6 / 10 = 0.6
Using the Pythagorean identity, we can find the sine ratio:
sin^2(A) = 1 - cos^2(A) = 1 - 0.6^2 = 1 - 0.36 = 0.64
Now, take the square root of both sides to find the sine ratio:
sin(A) = √0.64 = 0.8
Now that we have the sine ratio, we can find the length of the opposite side:
opposite side = sin(A) * hypotenuse = 0.8 * 10 = 8 cm
Problem 6-10: Solving Trigonometric Equations
These problems require you to solve trigonometric equations involving sine, cosine, and tangent functions. To solve these equations, you need to use the unit circle, trigonometric identities, and algebraic manipulations.- Use the unit circle to identify the reference angle and the quadrant of the given angle.
- Apply trigonometric identities to simplify the equation.
- Use algebraic manipulations to isolate the variable.
Since the sine function is positive in the first and second quadrants, we can use the inverse sine function to find the reference angle:
x = arcsin(0.7) = 0.85 (in radians)
Now, we need to find the angle in the interval [0, 2π). Since the angle is in the first quadrant, the value of x is 0.85 radians.
Problem 11-15: Trigonometric Applications
These problems apply trigonometric concepts to real-world situations, such as sound waves, light waves, and navigation. To solve these problems, you need to understand the relationships between angles, frequencies, and wavelengths.| Frequency (Hz) | Wavelength (m) | Speed of Sound (m/s) |
|---|---|---|
| 20 Hz | 17 m | 340 m/s |
| 100 Hz | 3.4 m | 340 m/s |
| 1000 Hz | 0.34 m | 340 m/s |
Using the formula v = fλ, we can find the speed of the sound wave:
v = fλ = 100 * 3.4 = 340 m/s
Problem 16-20: Trigonometric Graphs
These problems focus on graphing trigonometric functions, including sine, cosine, and tangent. To solve these problems, you need to understand the periodicity, amplitude, and phase shift of the functions.- Recall the general form of the sine and cosine functions: y = a sin(bx) and y = a cos(bx).
- Understand the effects of the amplitude, period, and phase shift on the graph.
- Use the unit circle to identify key points on the graph.
The graph of y = 2 cos(x) will have an amplitude of 2 and a period of 2π.
Using the unit circle, we can identify key points on the graph:
(0, 2), (π/2, 0), π, (3π/2, -2), and (2π, 2)
Problem 21-25: Trigonometric Identities
These problems require you to prove and apply various trigonometric identities, including the Pythagorean identity and the co-function identities.- Recall the Pythagorean identity: sin^2(A) + cos^2(A) = 1.
- Use the co-function identities to simplify expressions.
- Apply trigonometric identities to solve equations.
Using the co-function identity, we can rewrite the identity as:
sin(A) = cos(π/2 - A) = sin(π/2 - A) = cos(A)
Problem 26-30: Trigonometric Applications
These problems apply trigonometric concepts to real-world situations, such as navigation, sound waves, and light waves. To solve these problems, you need to understand the relationships between angles, frequencies, and wavelengths.| Angle (°) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 30° | 0.5 | 0.866 | 0.577 |
| 60° | 0.866 | 0.5 | 1.732 |
| 90° | 1 | 0 | undefined |
Using the cosine ratio, we can find the distance from the ship to the point directly in line with the shore:
cos(60°) = adjacent side / hypotenuse = x / 0.866
Now, solve for x:
x = 0.866 * cos(60°) = 0.866 * 0.5 = 0.433 km