Ratios and Graphs of Trigonometric Functions

Why A (0.5) is correct:
30° is a special angle with a known value: sin(30°) = 1/2 = 0.5. This comes from the properties of a 30-60-90 triangle, where the side opposite the 30° angle is always half the hypotenuse.

Why the others are wrong:
- B (0.866): This is sin(60°), not sin(30°). It's the reciprocal ratio in the 30-60-90 triangle.
- C (1): sin(90°) = 1, not sin(30°).
- D (0): sin(0°) = 0, not sin(30°).

Why A is correct:
When cos(θ) = 0.5, you're looking for angles where the x-coordinate on the unit circle equals 0.5. The cosine is positive in the first and fourth quadrants. At 60°, cos(60°) = 0.5, and at 300° (or –60°), cos(300°) = 0.5.

Why the others are wrong:
- B (60° and 240°): 60° is correct, but 240° is in the third quadrant where cosine is negative, so cos(240°) = –0.5, not 0.5.
- C (90° and 270°): These are where cosine equals 0, not 0.5.
- D (30° and 330°): cos(30°) = √3/2 ≈ 0.866 and cos(330°) = √3/2, both ≠ 0.5.

Correct answer: [-1, 1]

• Sine oscillates between a minimum of -1 and a maximum of 1 for any angle. At 270°, sin(x) = -1; at 90°, sin(x) = 1.
• Option B [0, 1]: Wrong because sine goes negative—it reaches -1 at 270°, not just 0.
• Option C [-2, 2]: Wrong because sine never exceeds ±1 in magnitude, no matter the angle.
• Option D [0, 2]: Wrong for the same reason as C—sine's maximum is 1, not 2, and it also goes negative.

Why A is correct:
tan(x) = sin(x)/cos(x), so vertical asymptotes occur where cos(x) = 0. In the range 0° to 360°, cosine equals zero at exactly 90° and 270°, making the function undefined there.

Why the others are wrong:
- B (0° and 180°): cos(0°) = 1 and cos(180°) = -1, so tan is defined at both points.
- C (45° and 135°): Both have defined tangent values (1 and -1 respectively).
- D (60° and 240°): Both have defined tangent values (√3 and √3/3 respectively).

Why A is correct:
The Pythagorean theorem states a² + b² = c². With legs of 3 and 4: 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

Why the others are wrong:
- B (6): This is just guessing a number larger than both legs; it doesn't follow from the theorem.
- C (7): Also just a guess; √25 ≠ 7.
- D (4): This is just one of the original leg lengths, not the hypotenuse.

Why A is correct:
The period is the horizontal distance before the sine function repeats its pattern. Starting at sin(0°) = 0, the function completes one full cycle and returns to the same value at sin(360°) = 0. This full cycle takes 360°.

Why the others are wrong:
- B (180°): At 180°, sin(180°) = 0, but the pattern hasn't fully repeated—you're only halfway through the cycle.
- C (90°): At 90°, sin(90°) = 1 (the peak), which is just a quarter of the cycle, not a complete repetition.
- D (45°): This is an arbitrary angle with no special significance to the sine function's repetition pattern.

Why A is correct:
At 180°, the point on the unit circle is at (-1, 0). Since tan = sin/cos = y/x, we get 0/(-1) = 0.

Why the others are wrong:
- B (1): This is tan(45°), not 180°
- C (0.5): No standard angle has a tangent of 0.5
- D (-1): This is tan(135°) or tan(-45°), not 180°

Why A is correct:
The cosine function has a maximum value of 1, which occurs when x = 0° and x = 360° (these are the same point on the unit circle, just one full rotation apart). At both angles, cos(x) = 1.

Why the others are wrong:
- B (x = 90°): cos(90°) = 0, not a maximum
- C (x = 180°): cos(180°) = –1, which is actually the *minimum* value
- D (x = 270°): cos(270°) = 0, not a maximum

# Explanation

Why A is correct:
Use the Pythagorean identity: sin²θ + cos²θ = 1. Since sin θ = 3/5, we get (3/5)² + cos²θ = 1, so 9/25 + cos²θ = 1. Solving: cos²θ = 16/25, therefore cos θ = 4/5.

Why the others are wrong:
- B (3/5): This is sin θ, not cos θ—they're different ratios.
- C (5/3): This is the reciprocal of sin θ (cosecant), not cosine.
- D (4/3): This would be tan θ (opposite/adjacent), not cos θ.

A. 1 ✓

The sine function reaches its highest point at 1, which occurs at x = 90°. You can verify this by remembering that sin(90°) = 1.

Why the others are wrong:
- B. 0 – This is where sine crosses the x-axis (at 0°, 180°, 360°), not the maximum.
- C. -1 – This is the *minimum* value, occurring at x = 270°.
- D. 1/2 – While sin(30°) = 1/2, this is just one value on the curve, not the highest point.

• Why A is correct: Use the Pythagorean identity: sin²θ + cos²θ = 1. Substitute cos θ = 5/13, then sin²θ = 1 - (5/13)² = 1 - 25/169 = 144/169. Taking the positive square root (since θ is in a right triangle), sin θ = 12/13.

• Why B is wrong: 5/12 is the ratio of adjacent to opposite sides, which equals cot θ or 1/tan θ—not sin θ.

• Why C is wrong: 13/12 is greater than 1, but sine values must always be between 0 and 1.

• Why D is wrong: This just repeats the cosine value; sine and cosine are different ratios.

• A. 3 is correct — The amplitude of a sine function y = a sin x is the absolute value of the number multiplying sin x. Here, that number is 3, so the amplitude is 3. This means the wave oscillates 3 units above and below the center line.

• B. 1 is wrong — This is the amplitude of y = sin x (with no multiplier), not y = 3 sin x.

• C. 2 is wrong — This doesn't relate to this function; it might confuse amplitude with period or another property.

• D. 1/2 is wrong — This is too small; it would only be correct if the function were y = (1/2) sin x.

Why A is correct:
At 0°, the angle points along the positive x-axis on the unit circle, where the x-coordinate equals 1. Since cosine gives the x-coordinate, cos 0° = 1.

Why the others are wrong:
- B (0): This is sin 0°, not cosine. Sine gives the y-coordinate, which is 0 at this angle.
- C (−1): This would be cos 180°, the opposite direction on the circle.
- D (1/2): This is cos 60°, a different angle entirely.

A. -1 is correct because cosine reaches its lowest point at x = 180°, where cos(180°) = -1. This is the minimum value the function achieves across the full range.

Why the others are wrong:
- B (0): Cosine equals 0 at x = 90° and x = 270°, but these are not minimum values—they're just points where the curve crosses the x-axis.
- C (1): This is actually the *maximum* value (at x = 0° and x = 360°), not the minimum.
- D (-1/2): While cosine does pass through -1/2 at certain angles, it goes lower than this, so it's not the minimum.

Why A is correct:
The amplitude is the distance from the middle of the wave to its peak (or trough). For y = 2 cos x, the coefficient 2 stretches the cosine function vertically, so the wave oscillates between –2 and +2. The amplitude is 2.

Why the others are wrong:
- B (1): That's the amplitude of plain cos x with no coefficient—this function has been stretched by a factor of 2.
- C (1/2): This is the reciprocal of the correct answer; it would mean the wave is *compressed*, not stretched.
- D (0): Amplitude can't be zero; that would mean no wave at all.

Why A is correct:
sin 90° = 1. At 90°, the angle points straight up on the unit circle, where the y-coordinate is at its maximum value of 1.

Why others are wrong:
- B (0): This is sin 0° or sin 180°—angles on the horizontal axis where the y-coordinate is zero.
- C (1/2): This is sin 30°, a specific acute angle, not 90°.
- D (–1): This is sin 270°, which points straight down where the y-coordinate is negative.

Why A is correct:
Tan θ = opposite/adjacent = 4/3, so use the Pythagorean theorem to find the hypotenuse: √(4² + 3²) = √25 = 5. Then sin θ = opposite/hypotenuse = 4/5.

Why the others are wrong:
- B (3/5): This is cos θ (adjacent/hypotenuse), not sin θ.
- C (4/3): This is tan θ itself, not sin θ.
- D (5/4): This is the reciprocal of sin θ (which is csc θ), not sin θ.

• A. 1 is correct. The negative sign flips the cosine curve upside down. Since cos x ranges from -1 to 1, negating it gives -cos x ranging from 1 to -1. The maximum (highest point) occurs when -cos x = 1, which happens at x = 180°.

• B. -1 is wrong. This is actually the *minimum* value of -cos x, not the maximum.

• C. 0 is wrong. While -cos x does equal 0 at certain points (90° and 270°), it's not the highest value the function reaches.

• D. -(1)/(2) is wrong. This is just one value that -cos x takes, but not an extreme (maximum or minimum).

Why A is correct:
cos θ = -1/2 is negative, so θ must be in Quadrant II (90°–180°) or Quadrant III (180°–270°). The reference angle is 60° (since cos 60° = 1/2). In Quadrant II: 180° − 60° = 120°. In Quadrant III: 180° + 60° = 240°.

Why the others are wrong:
- B (60°, 300°): These give cos 60° = 1/2 and cos 300° = 1/2 (both positive, not negative)
- C (90°, 270°): These give cos 90° = 0 and cos 270° = 0 (not -1/2)
- D (150°, 210°): These give cos 150° = -√3/2 and cos 210° = -√3/2 (close, but the wrong negative value)