Question

The point $ (7,24) $ is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Answer 1

Hint: In order to determine exact values of all six trigonometric function of the angle in the above question ,calculate $ r = \sqrt {{x^2} + {y^2}} $ where x will be 7 and y will be 24.And then find all the trigonometric ratios considering Hypotenuse as r ,opposite as 24 and adjacent as 7.
Formula:
\[
  \sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}} \\
  \cos \theta = \dfrac{{Adjacent}}{{{\text{Hypotenuse}}}} \\
  \tan \theta = \dfrac{{{\text{Opposite}}}}{{Adjacent}} \\
  \cos ec\theta = \dfrac{{{\text{Hypotenuse}}}}{{{\text{Opposite}}}} \\
  sec\theta = \dfrac{{{\text{Hypotenuse}}}}{{Adjacent}} \\
  \cot \theta = \dfrac{{Adjacent}}{{{\text{Opposite}}}} \\
 \]

Complete step-by-step answer:
Given a point P $ (7,24) $ which is on the terminal side of an angle in standard position.
Let x be 7 and y be 24

Calculating r using formula $ r = \sqrt {{x^2} + {y^2}} $
 $
  r = \sqrt {{{(7)}^2} + {{(24)}^2}} \\
  r = \sqrt {49 + 576} \\
  r = \sqrt {625} \\
  r = 25 \;
  $
Hence , value of r is $ 25 $
Therefore Calculating all the trigonometric ratios as
\[
  \sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}} = \dfrac{y}{r} = \dfrac{{24}}{{25}} \\
  \cos \theta = \dfrac{{Adjacent}}{{{\text{Hypotenuse}}}} = \dfrac{x}{r} = \dfrac{7}{{25}} \\
  \tan \theta = \dfrac{{{\text{Opposite}}}}{{Adjacent}} = \dfrac{y}{x} = \dfrac{{24}}{7} \\
  \cos ec\theta = \dfrac{{{\text{Hypotenuse}}}}{{{\text{Opposite}}}} = \dfrac{r}{y} = \dfrac{{25}}{{24}} \\
  sec\theta = \dfrac{{{\text{Hypotenuse}}}}{{Adjacent}} = \dfrac{r}{x} = \dfrac{{25}}{7} \\
  \cot \theta = \dfrac{{Adjacent}}{{{\text{Opposite}}}} = \dfrac{x}{y} = \dfrac{7}{{24}} \;
 \]

Note: 1. Periodic Function= A function $ f(x) $ is said to be a periodic function if there exists a real number T > 0 such that $ f(x + T) = f(x) $ for all x.
If T is the smallest positive real number such that $ f(x + T) = f(x) $ for all x, then T is called the fundamental period of $ f(x) $ .
Since $ \sin \,(2n\pi + \theta ) = \sin \theta $ for all values of $ \theta $ and n $ \in $ N.
2. Even Function – A function $ f(x) $ is said to be an even function ,if $ f( - x) = f(x) $ for all x in its domain.
Odd Function – A function $ f(x) $ is said to be an even function ,if $ f( - x) = - f(x) $ for all x in its domain.
We know that $ \sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta ) = - \tan \theta $
Therefore, $ \sin \theta $ and $ \tan \theta $ and their reciprocals, $ \cos ec\theta $ and $ \cot \theta $ are odd functions whereas \[\cos \theta \] and its reciprocal \[\sec \theta \] are even functions.