Trigonometric Ration of Any Angle

6.2   Trigonometric Ration of Any Angle
  • The diagram below shows the triangle \(ABC\)
  • The trigonometric ratios can be defined as follows:
    \(\text{sin} \ \theta = \dfrac{\text{opposite side}}{\text{hypotenuse}} = \dfrac{BC}{AB}\)    
  \(\text{kos} \ \theta = \dfrac{\text{adjacent side}}{\text{hypotenuse}} = \dfrac{AC}{AB}\)  
  \(\text{tan} \ \theta = \dfrac{\text{opposite side}}{\text{adjacent side}} = \dfrac{BC}{AC}\)  


  • There are three more ratios that are the reciprocals of these trigonometric ratios:
   \(\text{cosec }\theta= \dfrac{1}{\text{sin }\theta}\)   
   \(\text{sec }\theta= \dfrac{1}{\text{cos }\theta}\)   
   \(\text{cot }\theta= \dfrac{1}{\text{tan }\theta}\)   
  • The angles \(A\) and \(B\) are complementary angles to each other if \(A + B = 90^{\circ}\). Hence,

   \(A=90^{\circ} -B \text{ and }B=90^{\circ}-A\)   
  • The formulae of the complementary angles are as follows:
  \(\text{sin }\theta= \text{cos } (90^{\circ}-\theta)\)    \(\text{cos }\theta= \text{sin } (90^{\circ}-\theta)\)    \(\text{tan }\theta= \text{cot } (90^{\circ}-\theta)\)  
  \(\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)\)   \(\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)\)   \(\text{cot }\theta= \text{tan } (90^{\circ}-\theta)\)  

Given that \(\text{sin } 77^{\circ} = 0.9744\) and \(\text{cos } 77^{\circ} = 0.225\).

Find the value of \(\text{cos }13^{\circ}\).


Given \(\text{cos }63^{\circ} = k\), where \(k \text{ > }0\).

Find the value of \(\text{cosec }27^{\circ}\) in terms of \(k\).

(a) \(\begin{aligned} \text{cos }\theta&= \text{sin } (90^{\circ}-\theta)\\ \text{cos }13^{\circ}&= \text{sin } (90^{\circ}-13^{\circ})\\ &= \text{sin }77^{\circ}\\ &= 0.9744 \end{aligned}\)
(b) \(\begin{aligned} \text{cosec }\theta&= \text{sec } (90^{\circ}-\theta)\\ \text{cosec }27^{\circ}&= \text{sec } (90^{\circ}-27^{\circ})\\ &= \text{sec }63^{\circ}\\ &= \dfrac{1}{\text{cos }63^{\circ}}\\ &= \dfrac{1}{k} \end{aligned}\)
  • \(4\) methods to determine the values of the trigonometric rations for any angle:


Method 1: Use a calculator

  • The values of sine, cosine and tangent can be determined by using a calculator

  • The values for cosecant, secant and cotangent can be calculated by inversing the values of the trigonometric ratios of sine, cosine and tangent


Method 2: Use a unit circle


Use the unit circle above and state the value of \(\text{cos }135^{\circ}\).


The coordinates that correspond to \(135^{\circ}\) are

\(\begin{pmatrix} -\dfrac{1}{\sqrt2}, \ \dfrac{1}{\sqrt2} \end{pmatrix}\) and \(\text{cos }135^{\circ} = x\text{-coordinate}\).

Hence, \(\text{cos }135^{\circ} = -\dfrac{1}{\sqrt2}\).


Method 3: Use the corresponding trigonometric ratio of the reference angle

  • The diagram shows the reference angles, \(\alpha\) for the angles \( 0° \leqslant \theta \leqslant 360^{\circ}\) or \( 0° \leqslant \theta \leqslant 2\pi\)


Method 4: Use a right-angled triangle

  • The trigonometric ratios of special angles \(30^{\circ}, 45^{\circ} \text{ and }60^{\circ}\) can be determined by using right-angled triangles

Angle \ Ratio sin cos tan cosec sec cot
\(30^{\circ}\) \(\dfrac{\pi}{6}\) \(\dfrac{1}{2}\) \(\dfrac{\sqrt3}{2}\) \(\dfrac{1}{\sqrt3}\) \(2\) \(\dfrac{2}{\sqrt3}\) \(\sqrt3\)
\(45^{\circ}\) \(\dfrac{\pi}{4}\) \(\dfrac{1}{\sqrt2}\) \(\dfrac{1}{\sqrt2}\) \(1\) \(\sqrt2\) \(\sqrt2\) \(1\)
\(60^{\circ}\) \(\dfrac{\pi}{3}\) \(\dfrac{\sqrt3}{2}\) \(\dfrac{1}{2}\) \(\sqrt3\) \(\dfrac{2}{\sqrt3}\) \(2\) \(\dfrac{1}{\sqrt3}\)