Q1 Exercise 4.2A: Trigonometry 1 David Rayner Complete Mathematics Extended Book Solution

You will need a protractor for this exercise. 1. Draw a circle of radius 10 cm on plain paper and construct a tangent to touch the circle at T. To construct the tangent, first draw a radius, label the point T where the radius meets the circle, then use a protractor to construct a right angle at T. Draw OA, OB and OC such that AOT = 20° BOT = 40° COT = 50° Measure the length AT and compare it with the value for tan 20° given on a calculator. Repeat for BT, CT and for other angles of your own choice. Discuss with a partner what you discover.



🔧 You Will Need:

  • Plain paper

  • Compass

  • Ruler

  • Protractor

  • Pencil

  • Calculator

✅ Step-by-Step Construction and Calculations

Step 1: Draw the Circle

  • Use your compass to draw a circle of radius 10 cm.

  • Mark the center as O.

Step 2: Draw a Radius

  • Pick any point on the circle and label it T.

  • Draw a radius OT = 10 cm.

Step 3: Construct the Tangent at T

  • At point T, use a protractor to construct a 90° angle to radius OT.

  • Draw a straight line at this 90° — this is the tangent at point T.

Step 4: Draw Angles AOT = 20°, BOT = 40°, COT = 50°

  • Using your protractor, draw angles:

    • From radius OT, mark a 20° angle and draw a line from O — label this point on the tangent line as A.

    • From OT, mark a 40° angle and draw a line from O — label this point as B.

    • From OT, mark a 50° angle and draw a line from O — label this point as C.

You should now have triangle segments OA, OB, and OC with angles at O being 20°, 40°, and 50°, all meeting the tangent line at T.

Step 5: Measure Lengths

  • Use a ruler to measure:

    • AT

    • BT

    • CT

Step 6: Compare With Trigonometric Values

We will compare the measured lengths with theoretical values from trigonometry.

Since OT = 10 cm, and we know:

tan⁡(θ)=oppositeadjacent=length from T to the point on tangentOT​

So,

  • For ∠AOT = 20°:

    tan⁡(20°)≈0.3640AT=OT×tan⁡(20°)=10×0.3640=3.64 cm\tan(20°) ≈ 0.3640 \\ AT = OT × \tan(20°) = 10 × 0.3640 = 3.64 \text{ cm}

  • For ∠BOT = 40°:

    tan⁡(40°)≈0.8391BT=10×0.8391=8.39 cm\tan(40°) ≈ 0.8391 \\ BT = 10 × 0.8391 = 8.39 \text{ cm}

  • For ∠COT = 50°:

    tan⁡(50°)≈1.1918CT=10×1.1918=11.92 cm\tan(50°) ≈ 1.1918 \\ CT = 10 × 1.1918 = 11.92 \text{ cm}

Compare these calculated lengths with your measured lengths of AT, BT, and CT.

📈 What Do You Notice?

  • As the angle increases, the tangent length also increases.

  • The measured lengths closely match the calculated values using the tangent function.

  • This activity demonstrates that:

Length from point on tangent (e.g., AT)=radius×tan⁡(angle at center)\text{Length from point on tangent (e.g., AT)} = \text{radius} × \tan(\text{angle at center})

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