Use Trigonometric Identity cos(A+B)

Example how to use the identity cos(A+B) corresponding to question 2 in trigonometry_2.

Example 1


If $A$ and $B$ are angles in quadrant I and $\cos A = \dfrac{1}{7}$ and $\cos B = \dfrac{2}{5}$, then $\cos(A+B)=$

  1. $\dfrac{19}{35}$

  2. $\dfrac{2}{35}$

  3. $\dfrac{2-12\sqrt 7}{35}$

  4. $\dfrac{2+12\sqrt 7}{35}$

  5. $\dfrac{-10\sqrt 7}{35}$

Solution


  1. We first expand $\cos(A+B)=$ as follows

    $\cos(A+B)= \cos A \cos B - \sin A \sin B$

  2. We are given $\cos A = \dfrac{1}{7}$ and $\cos B = \dfrac{2}{5}$ and we now need to find $\sin A$ and $\sin B$

    $\sin A=\sqrt{1-cos^2 A}=\sqrt{1-\dfrac{1}{49}}=\sqrt{\dfrac{49-1}{49}}=\dfrac{\sqrt{48}}{7}=\dfrac{\sqrt{3 \cdot 16}}{7}=\dfrac{4 \sqrt 3}{7}$


    3. $\sin B=\sqrt{1-cos^2 B}=\sqrt{1-\dfrac{4}{25}}=\sqrt{\dfrac{25-4}{25}}=\dfrac{\sqrt{21}}{5}$

  3. We now substitute $\sin A$ and $\sin B$ by their values found above in the main indentity and evaluate $\cos(A+B)$ as follows

    $\cos(A+B)= \cos A \cos B - \sin A \sin B = \dfrac {1}{7} \cdot \dfrac {2}{5}- \dfrac{4 \sqrt 3}{7} \cdot \dfrac{\sqrt{21}}{5}$

    $=\dfrac{2-4\sqrt 3\sqrt{3 \cdot 7}}{35} = \dfrac{2-4 \cdot 3\sqrt 7}{35} = = \dfrac{2-12\sqrt 7}{35} $

    Answer C