Trigonometry - Flashcards
Master trigonometry concepts through problem-solving practice! These flashcards focus on applying trigonometric functions, identities, and theorems to solve real problems. Each card presents a practical problem with step-by-step solutions to build your problem-solving skills for UPCAT and other college entrance exams.
Flashcards
Find sin θ, cos θ, and tan θ if the point (-3, 4) lies on the terminal side of angle θ in standard position.
Answer
Step 1: Find r using distance formula: r = √(x² + y²) = √((-3)² + 4²) = √(9 + 16) = √25 = 5
Step 2: Apply definitions:
- sin θ = y/r = 4/5
- cos θ = x/r = -3/5
- tan θ = y/x = 4/(-3) = -4/3
Answer: sin θ = 4/5, cos θ = -3/5, tan θ = -4/3
Convert 135° to radians.
Answer
Step 1: Use conversion formula: radians = degrees × (π/180°)
Step 2: Substitute: 135° × (π/180°)
Step 3: Simplify: (135π)/180 = (3π)/4
Answer: 3π/4 radians
A ladder 10 meters long leans against a wall, making a 60° angle with the ground. How high up the wall does the ladder reach?
Answer
Step 1: Identify the right triangle - ladder is hypotenuse (10m), height is opposite to 60° angle
Step 2: Use sine function: sin 60° = opposite/hypotenuse
Step 3: sin 60° = height/10
Step 4: height = 10 × sin 60° = 10 × (√3/2) = 5√3
Step 5: Calculate: 5√3 ≈ 8.66 meters
Answer: 5√3 or approximately 8.66 meters
Find the exact value of cos 240°.
Answer
Step 1: Determine the quadrant - 240° is in Quadrant III (180° < 240° < 270°)
Step 2: Find reference angle: 240° - 180° = 60°
Step 3: In Quadrant III, cosine is negative
Step 4: cos 240° = -cos 60° = -1/2
Answer: -1/2
If sin θ = 3/5 and θ is in Quadrant II, find cos θ and tan θ.
Answer
Step 1: Use Pythagorean identity: sin²θ + cos²θ = 1
Step 2: (3/5)² + cos²θ = 1
Step 3: 9/25 + cos²θ = 1
Step 4: cos²θ = 1 - 9/25 = 16/25
Step 5: cos θ = ±4/5, but in Quadrant II, cosine is negative, so cos θ = -4/5
Step 6: tan θ = sin θ/cos θ = (3/5)/(-4/5) = -3/4
Answer: cos θ = -4/5, tan θ = -3/4
Find the arc length of a circle with radius 8 cm if the central angle is 45°.
Answer
Step 1: Convert angle to radians: 45° × (π/180°) = π/4 radians
Step 2: Use arc length formula: s = rθ (where θ is in radians)
Step 3: s = 8 × (π/4) = 2π cm
Step 4: Calculate decimal: 2π ≈ 6.28 cm
Answer: 2π cm or approximately 6.28 cm
In a right triangle, if the opposite side is 7 and the hypotenuse is 25, find sin θ, cos θ, and tan θ.
Answer
Step 1: Find adjacent side using Pythagorean theorem: a² + 7² = 25²
Step 2: a² + 49 = 625, so a² = 576, therefore a = 24
Step 3: Apply trigonometric ratios:
- sin θ = opposite/hypotenuse = 7/25
- cos θ = adjacent/hypotenuse = 24/25
- tan θ = opposite/adjacent = 7/24
Answer: sin θ = 7/25, cos θ = 24/25, tan θ = 7/24
Simplify: sin²θ + cos²θ + tan²θ - sec²θ
Answer
Step 1: Use fundamental identities: sin²θ + cos²θ = 1 and sec²θ - tan²θ = 1
Step 2: Rearrange the second identity: tan²θ - sec²θ = -1
Step 3: Substitute: (sin²θ + cos²θ) + (tan²θ - sec²θ)
Step 4: = 1 + (-1) = 0
Answer: 0
Find the coterminal angle between 0° and 360° for the angle 890°.
Answer
Step 1: Since 890° > 360°, subtract multiples of 360°
Step 2: 890° ÷ 360° = 2.47..., so we subtract 2 × 360° = 720°
Step 3: 890° - 720° = 170°
Step 4: Check: 0° ≤ 170° ≤ 360° ✓
Answer: 170°
A tree casts a shadow 15 meters long when the angle of elevation of the sun is 35°. Find the height of the tree.
Answer
Step 1: Draw right triangle - height is opposite to 35°, shadow is adjacent
Step 2: Use tangent function: tan 35° = height/shadow
Step 3: tan 35° = height/15
Step 4: height = 15 × tan 35°
Step 5: height ≈ 15 × 0.7002 ≈ 10.5 meters
Answer: Approximately 10.5 meters
Convert 5π/6 radians to degrees.
Answer
Step 1: Use conversion formula: degrees = radians × (180°/π)
Step 2: (5π/6) × (180°/π)
Step 3: Simplify: (5 × 180°)/6 = 900°/6 = 150°
Answer: 150°
If csc θ = 13/5, find sin θ and determine which quadrants θ could be in.
Answer
Step 1: Use reciprocal identity: csc θ = 1/sin θ
Step 2: 13/5 = 1/sin θ, so sin θ = 5/13
Step 3: Since sin θ > 0, θ must be in Quadrant I or II
Step 4: Verify: csc θ = 1/(5/13) = 13/5 ✓
Answer: sin θ = 5/13; θ is in Quadrant I or II
In triangle ABC, a = 8, b = 6, and C = 60°. Find side c using the Law of Cosines.
Answer
Step 1: Apply Law of Cosines: c² = a² + b² - 2ab cos C
Step 2: c² = 8² + 6² - 2(8)(6)cos 60°
Step 3: c² = 64 + 36 - 96 cos 60°
Step 4: c² = 100 - 96(1/2) = 100 - 48 = 52
Step 5: c = √52 = √(4×13) = 2√13
Step 6: c ≈ 7.21
Answer: c = 2√13 or approximately 7.21
Find all values of θ between 0° and 360° where sin θ = -√2/2.
Answer
Step 1: Recognize that |sin θ| = √2/2 corresponds to reference angle 45°
Step 2: Since sin θ < 0, θ must be in Quadrant III or IV
Step 3: For Quadrant III: θ = 180° + 45° = 225°
Step 4: For Quadrant IV: θ = 360° - 45° = 315°
Step 5: Verify: sin 225° = sin 315° = -√2/2 ✓
Answer: θ = 225° and θ = 315°
A wheel with radius 4 inches rotates through an angle of 2π/3 radians. How far does a point on the rim travel?
Answer
Step 1: This is an arc length problem where the point travels along the arc
Step 2: Use arc length formula: s = rθ
Step 3: s = 4 × (2π/3) = 8π/3 inches
Step 4: Calculate decimal: 8π/3 ≈ 8.38 inches
Answer: 8π/3 inches or approximately 8.38 inches
If tan θ = -3/4 and θ is in Quadrant II, find sin θ and cos θ.
Answer
Step 1: In Quadrant II, sin θ > 0 and cos θ < 0
Step 2: Use identity: tan²θ + 1 = sec²θ
Step 3: (-3/4)² + 1 = sec²θ, so 9/16 + 1 = sec²θ
Step 4: sec²θ = 25/16, so sec θ = ±5/4
Step 5: In Quadrant II, sec θ < 0, so sec θ = -5/4
Step 6: cos θ = 1/sec θ = -4/5
Step 7: sin θ = tan θ × cos θ = (-3/4) × (-4/5) = 3/5
Answer: sin θ = 3/5, cos θ = -4/5
Find the exact value of sin 30° + cos 60° - tan 45°.
Answer
Step 1: Recall special angle values:
- sin 30° = 1/2
- cos 60° = 1/2
- tan 45° = 1
Step 2: Substitute: 1/2 + 1/2 - 1
Step 3: Calculate: 1 - 1 = 0
Answer: 0
In triangle PQR, P = 45°, Q = 75°, and side p = 10. Find side q using the Law of Sines.
Answer
Step 1: First find angle R: R = 180° - 45° - 75° = 60°
Step 2: Apply Law of Sines: p/sin P = q/sin Q
Step 3: 10/sin 45° = q/sin 75°
Step 4: q = 10 × (sin 75°/sin 45°)
Step 5: sin 45° = √2/2 ≈ 0.707, sin 75° ≈ 0.966
Step 6: q = 10 × (0.966/0.707) ≈ 13.66
Answer: q ≈ 13.66
Verify the identity: (1 + cos θ)(1 - cos θ) = sin²θ
Answer
Step 1: Expand left side using difference of squares: (1 + cos θ)(1 - cos θ) = 1² - (cos θ)²
Step 2: Simplify: = 1 - cos²θ
Step 3: Use Pythagorean identity: sin²θ + cos²θ = 1
Step 4: Rearrange: sin²θ = 1 - cos²θ
Step 5: Substitute: 1 - cos²θ = sin²θ ✓
Answer: Identity verified. Both sides equal sin²θ.
A surveyor measures the angle of elevation to the top of a building as 25° from a point 50 meters away. Find the height of the building.
Answer
Step 1: Draw right triangle - building height is opposite to 25° angle, 50m is adjacent
Step 2: Use tangent function: tan 25° = height/50
Step 3: height = 50 × tan 25°
Step 4: tan 25° ≈ 0.466
Step 5: height = 50 × 0.466 = 23.3 meters
Answer: Approximately 23.3 meters
Diagrams
Trigonometry Concept Overview
Complete overview of trigonometry topics showing the relationships between different concepts and their applications.
Problem-Solving Strategy for Trigonometry
Decision tree for choosing the right trigonometric approach based on what information is given in the problem.
UPCAT Trigonometry Question Distribution
Typical distribution of trigonometry question types in UPCAT and similar college entrance exams, helping students focus their study time effectively.