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Finding an angle of a right-angled triangle

The plane and space, grade 11 (3 units) Lesson 10 of 15Learning

Finding an angle of a right-angled triangle

From the ratio to the angle

When two sides are known, so is the ratio between them – and the ratio gives the angle.

  • If tan⁡α=0.75\displaystyle {\tan\alpha}={0.75}, then α\displaystyle \alpha is the angle whose tangent is 0.75\displaystyle 0.75. We write α=tan⁡−10.75\displaystyle {\alpha}={\tan^{-1}0.75}.
  • On a calculator: SHIFT, then tan. tan⁡−10.75≈36.87∘\displaystyle \tan^{-1}0.75\approx36.87^\circ.
  • Likewise sin⁡−1\displaystyle \sin^{-1} and cos⁡−1\displaystyle \cos^{-1}: sin⁡−10.5=30∘\displaystyle {\sin^{-1}0.5}={30^\circ}, because sin⁡30∘=0.5\displaystyle {\sin30^\circ}={0.5}.

With a\displaystyle a the side opposite the angle, b\displaystyle b the side adjacent to it and c\displaystyle c the hypotenuse:

α\displaystyle \alpha =\displaystyle = tan⁡−1ab\displaystyle \tan^{-1}\frac{a}{b}
α\displaystyle \alpha =\displaystyle = sin⁡−1ac\displaystyle \sin^{-1}\frac{a}{c}
α\displaystyle \alpha =\displaystyle = cos⁡−1bc\displaystyle \cos^{-1}\frac{b}{c}
Exercise 1

Find the angle, in degrees. Round to two decimal places.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Example: the angle of a ramp

        A ramp rises one metre over 12 m measured horizontally.
        What angle α\displaystyle \alpha does it make with the ground?

        • The height is opposite the angle, and the horizontal length is next to it: tan⁡α=112\displaystyle {\tan\alpha}={\frac{1}{12}}.
        • α=tan⁡−1112≈4.76∘\displaystyle {\alpha}={\tan^{-1}\frac{1}{12}\approx4.76^\circ}.
        Exercise 2

        A ladder 4 m long leans against a wall, its feet 1.2 m from the wall.
        What is the angle between the ladder and the ground? Round to two decimal places.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 3

          An upright pole 2 m tall casts a shadow 3 m long.
          At what angle do the sun’s rays meet the ground? Round to two decimal places.

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Exercise 4

            The legs of a right-angled triangle are 7 and 24.
            Find its two acute angles, and check that they add up to 90∘\displaystyle 90^\circ.
            Round to two decimal places.

            a
            The angle opposite the leg 7

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b
              The angle opposite the leg 24

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 5

                A ladder is best set at about 75∘\displaystyle 75^\circ to the ground.
                A ladder 3 m long stands with its feet 0.6 m from the wall.
                What is the angle between the ladder and the ground? Is it close to the recommended angle?

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power