Maths with David

    IBDP. Core. HL. Trigonometry. Trigonometric Ratios

    Opening Problem

    • The right angled triangles above both contain a 30 ^{ \circ } angle. From the information we are given, can we determine:
      • The lengths PQ and QR
      • The ratio of length PQ to QR?
    • The right angled triangles above both contain a 35 ^{\circ} angle. Can you explain why:
      • \frac{AB}{BC} = \frac{PQ}{QR}
      • Any other right angled triangle containing a 35 ^{ \circ } will have corresponding sides in the same ratio.

    Trigonometry is the study of the relationships between the side lengths and angles of triangles.

    • In the triangle above,
      • The hypotenuse (HYP) is the side opposite the right angle. It is the longest side of the triangle.
      • BC is the side opposite (OPP) angle \theta .
      • AB is the side adjacent (ADJ) to angle \theta .

    The trigonometric ratios for the angle \theta are sin \theta = \frac{OPP}{HYP} , cos \theta = \frac{ADJ}{HYP} , tan \theta = \frac{OPP}{ADJ} . They stand for sine, cosine and tangent.

    Notice that \frac{ sin \theta }{ cos \theta } = \frac{ \frac{OPP}{HYP} }{ \frac{ADJ}{HYP} } = \frac{OPP}{ADJ} = tan \theta .

    We can find the trigonometric ratios for any angle using a calculator.

    Worked Example 1

    • For the following triangle, find:
      • sin \theta
      • cos \theta
      • tan \theta

    Worked Example 2

    Find, correct to 3 significant figures, the unknown length in the following triangles:

    Exercise

    Answers