Radians - Y2

How to convert degrees to radians?

Step 1: Write down the angle in degrees. Step 2: Multiply by π/180. Step 3: Simplify the fraction. Step 4: Leave the answer in terms of π (do not use decimals unless asked). Remember: 180° = π radians, so multiplying by π/180 converts degrees to radians.

How to convert radians to degrees?

Step 1: Write down the angle in radians. Step 2: Multiply by 180/π. Step 3: Simplify and calculate if needed. Step 4: Add the degree symbol to your final answer. Remember: π radians = 180°.

How to remember key exact radian values?

Step 1: Memorise common conversions: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360°. Step 2: Use these to work out others by adding or subtracting. Step 3: Recognise that radian measure is based on π. Remember: most exam answers should be left in terms of π.

How to find arc length using radians?

Step 1: Make sure the angle θ is in radians (convert if needed). Step 2: Use the formula l = rθ. Step 3: Substitute the radius and angle carefully. Step 4: Multiply to find the arc length. Step 5: Include correct units. Remember: this formula only works when θ is in radians.

How to find the area of a sector using radians?

Step 1: Ensure θ is in radians. Step 2: Use the formula A = 1/2 r²θ. Step 3: Substitute radius and angle. Step 4: Multiply carefully. Step 5: Include square units. Remember: this is derived from the area of a full circle πr².

How to find the area of a segment?

Step 1: Find the area of the sector using A = 1/2 r²θ. Step 2: Find the area of the triangle using A = 1/2 r² sinθ. Step 3: Subtract triangle area from sector area. Step 4: Simplify fully. Remember: segment = sector − triangle.

How to solve trigonometric equations in radians?

Step 1: Make sure your calculator is in radian mode. Step 2: Rearrange the equation to isolate the trig function. Step 3: Use inverse trig to find the principal value. Step 4: Use quadrant rules (CAST) to find other solutions. Step 5: Add the correct period (2π for sin and cos, π for tan). Step 6: Write all solutions in the given interval.

How to solve equations like tan(2x) = k in radians?

Step 1: Isolate tan(2x). Step 2: Use inverse tan to find principal value for 2x. Step 3: Add π for the second solution since tan repeats every π. Step 4: Solve for x by dividing all answers by 2. Step 5: Check answers lie within the required interval.

How to use small angle approximations?

Step 1: Check that θ is close to 0 and measured in radians. Step 2: Use approximations: sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2. Step 3: Substitute into the expression. Step 4: Simplify algebraically. Remember: these only work when θ is small and in radians.

How to simplify expressions using small angle approximations?

Step 1: Replace sinθ with θ, tanθ with θ, and cosθ with 1 − θ²/2. Step 2: Expand brackets carefully. Step 3: Simplify like normal algebra. Step 4: Ignore higher powers if instructed. Remember: always state that θ is small and measured in radians.

How to show two areas are equal in radian problems?

Step 1: Write expressions for both areas separately (sector, triangle, rectangle etc). Step 2: Use correct radian formulas (l = rθ, A = 1/2 r²θ, triangle = 1/2 ab sinC). Step 3: Equate the two expressions. Step 4: Simplify step by step. Step 5: Rearrange to reach required result. Remember: show all algebra clearly.

How to find perimeter involving arcs?

Step 1: Identify straight edges and curved edges separately. Step 2: For curved edges use arc length formula l = rθ. Step 3: Add all straight sides and arc lengths together. Step 4: Simplify fully. Remember: arc length only works if θ is in radians.

How to apply radians in exam-style geometry problems?

Step 1: Label the diagram clearly. Step 2: Identify which formulas are needed (arc length, sector area, triangle area). Step 3: Substitute carefully using radians. Step 4: Simplify algebra step by step. Step 5: Keep answers exact unless told otherwise.

How to sketch trig graphs in radians?

Step 1: Replace degree key points with radian equivalents (0, π/2, π, 3π/2, 2π). Step 2: Sketch sine starting at (0,0). Step 3: Sketch cosine starting at (0,1). Step 4: Sketch tangent with asymptotes at π/2, 3π/2. Step 5: Label key intercepts clearly in terms of π.

How to remember trig graph periods in radians?

Step 1: Sine period = 2π. Step 2: Cosine period = 2π. Step 3: Tangent period = π. Step 4: If function is sin(nx), new period = 2π/n. Step 5: If tan(nx), new period = π/n. Remember: multiplying inside the bracket changes the period.