Trigonometric Functions

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We know that 180° = π radians. So, 1° = π/180 radians. Therefore, 45° = 45 × (π/180) = π/4 radians. Answer: π/4 radians.

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We know that π radians = 180°. So, 1 radian = 180°/π. Therefore, (3π/5) radians = (3π/5) × (180°/π) = (3 × 180°)/5 = 108°. Answer: 108°.

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We use the formula: l = rθ, where l = arc length, r = radius, θ = angle in radians. Given: l = 12 cm, θ = 0.8 rad. So, r = l / θ = 12 / 0.8 = 15 cm. Answer: 15 cm.

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We know that 180° = π radians. So, 1° = π/180 radians. Therefore, 120° = 120 × (π/180) = (2π)/3 radians. Answer: 2π/3 radians.

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Use the formula: l = rθ. Given: r = 9 cm, θ = π/3 rad. So, l = 9 × (π/3) = 3π cm. Answer: 3π cm (≈9.42 cm).

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5π/6 lies in the second quadrant (between π/2 and π). In the second quadrant, sine is positive. Therefore, sin(5π/6) is positive. Answer: Positive.

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4π/3 lies in the third quadrant (between π and 3π/2). In the third quadrant, tangent is positive (since both sine and cosine are negative). Therefore, tan(4π/3) is positive. Answer: Positive.

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We know: sin(π/3) = √3/2, cos(π/6) = √3/2, cos(π/3) = 1/2, sin(π/6) = 1/2. So, (√3/2)(√3/2) – (1/2)(1/2) = (3/4) – (1/4) = 2/4 = 1/2. Alternatively, this is sin(A–B) = sin(π/3 – π/6) = sin(π/6) = 1/2.