USTET Mathematics — TrigonometryFlash Cards

Find sin θ, cos θ, and tan θ if the point (-3, 4) lies on the terminal side of angle θ in standard position.

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.

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?

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°.

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.

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°.

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.

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

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°.

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.

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.

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.

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.

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.

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?

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.

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°.

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.

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

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.

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