Evaluate a Trigonometric Function form an Equation

Example on how to find the value of a trigonometric function from an equation. This corresponds to question 9 in trigonometry_2.

Example 1


What is $\sin x$ if $2 \tan x + 2 = 5$ and $0 \lt x \lt \dfrac{\pi}{2}$
  1. $\dfrac{2}{\sqrt {13}}$
  2. $\dfrac{3}{\sqrt {13}}$
  3. $-\dfrac{3}{\sqrt {13}}$
  4. $\dfrac{2}{3}$
  5. $\dfrac{4}{\sqrt {13}}$

Solution


  1. We first use the given equation to find $\tan \theta$.

    $2 \tan x + 2 = 5$

    $ 2\tan x = 5-2$

    $ \tan x = \dfrac{3}{2} $

  2. We now use the definition of $\tan x$ using a point with coordinates $(a,b)$ on the terminal side of $x$

    $\tan x = \dfrac{3}{2} = \dfrac{b}{a}$


  3. Since $x$ is in quadrant $I$, $a$ and $b$ are positive. Hence

    $a = 2$ and $b = 3$

  4. Distance r from the origin to the point with coordinates (a,b) is given by

    $r = \sqrt{a^2+b^2}=\sqrt{4+9}=\sqrt{13}$

    $\sin x= \dfrac{b}{r} = \dfrac{3}{\sqrt {13}}$

    Answer B