Math Doubts

Find the value of $\cos{40°}$ $+$ $\cos{80°}$ $+$ $\cos{160°}$

$\cos{40^\circ}$ $+$ $\cos{80^\circ}$ $+$ $\cos{160^\circ}$

The values of cosine of these three angles are unknown. So, it cannot solved directly by substituting values of them in this trigonometric expression. So, an alternative approach should be used to simplify it.

Try sum to product transforming identity

All three terms are cosine functions. So, there is no problem to apply sum to product transformation identities.

$\cos{\alpha}+\cos{\beta}$ $=$ $2\cos{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\cos{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

In this math problem, $\alpha = 40^\circ$ and $\beta = 80^\circ$. Substitute them and simplify the trigonometric expression.

$=$ $2\cos{\Bigg(\dfrac{40^\circ + 80^\circ}{2}\Bigg)}\cos{\Bigg(\dfrac{40^\circ – 80^\circ}{2}\Bigg)}$ $+$ $\cos{160^\circ}$

$=$ $2\cos{\Bigg(\dfrac{120^\circ}{2}\Bigg)}\cos{\Bigg(\dfrac{-40^\circ}{2}\Bigg)}$ $+$ $\cos{160^\circ}$

$=$ $2\cos{60^\circ}\cos{(-20^\circ)}$ $+$ $\cos{160^\circ}$

Dealing cosine of negative angle

As per cos of negative angle identity, cosine of a negative angle is equal to the cosine of same positive angle.

$=$ $2\cos{60^\circ}\cos{20^\circ}$ $+$ $\cos{160^\circ}$

Simplifying the trigonometric expression

$=$ $2 \times \dfrac{1}{2} \times \cos{20^\circ}$ $+$ $\cos{160^\circ}$

The value of cosine of $60^\circ$ is equal to $\dfrac{1}{2}$. Replace $\cos{60^\circ}$ by this value to go ahead in simplifying this trigonometry problem.

$=$ $\dfrac{2}{2} \times \cos{20^\circ}$ $+$ $\cos{160^\circ}$

$=$ $\require{cancel} \dfrac{\cancel{2}}{\cancel{2}} \times \cos{20^\circ}$ $+$ $\cos{160^\circ}$

$=$ $\cos{20^\circ}$ $+$ $\cos{160^\circ}$

Sum to product transforming trigonometric identity can be used one more time but it can also be solved mathematically by trigonometric ratios of allied angles.

$=$ $\cos{20^\circ}$ $+$ $\cos{(180^\circ -20^\circ)}$

$=$ $\cos{20^\circ}$ $-\cos{20^\circ}$

$= 0$

$\,\,\, \therefore \,\,\,\,\,\,$ $\cos{40^\circ}$ $+$ $\cos{80^\circ}$ $+$ $\cos{160^\circ} = 0$

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more