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Question

Question: How do you solve\[\cos \theta = - 0.5?\]...

How do you solvecosθ=0.5?\cos \theta = - 0.5?

Explanation

Solution

we need to know trigonometric table values and the basic equations with the involvement
ofcosθ\cos \theta . To solve the given equation we need to find the value ofθ\theta . Also, it
involves the operation of addition/ subtraction/ multiplication/division. Also, we need to know how
to calculate cos1{\cos ^{ - 1}}the value in the scientific calculator.

Complete step by step solution:

The given question is shown below,
cosθ=0.5\cos \theta = - 0.5
We need to find the value θ\theta from the above equation. Before that, we need to know the
basic definition ofcosθ\cos \theta

The above figure represents a triangle marked with the opposite side, adjacent side, and hypotenuse
side according to the position ofθ\theta .
We know that,
\cos \theta = \dfrac{{adjacant}}{{hypotenuse}}$$$$ \to \left( 1 \right)
The given equation is,
\cos \theta = - 0.5$$$$ \to \left( 2 \right)
The above equation can also be written as,
\cos \theta = \dfrac{{ - 1}}{2}$$$$ \to \left( 3 \right)
By comparing the equation (1)\left( 1 \right)and(3)\left( 3 \right), we get the value of the adjacent
side is1 - 1 and the value of the hypotenuse side is22.
When the term cos\cos is the move from the left side to the right side of the equation it converts
intoarccos\arccos . Let’s find the value of θ\theta from the equation(3)\left( 3 \right), we get
(3)cosθ=12\left( 3 \right) \to \cos \theta = \dfrac{{ - 1}}{2}
θ=arccos(12)\theta = \arccos \left( {\dfrac{{ - 1}}{2}} \right)
By, using the trigonometric table value, we get
cos(60)=12\cos \left( {{{60}^ \circ }} \right) = \dfrac{1}{2}
cos(120)=12cos\left( {{{120}^ \circ }} \right) = \dfrac{{ - 1}}{2}
So we get,
\arccos \left( {\dfrac{{ - 1}}{2}} \right) = {120^ \circ }$$$$ \to \left( 4 \right)

So, the value ofθ\theta becomes,
θ=arccos(0.5)0.5\theta= arccos(-0.5)0.5

θ=120\theta = {120^ \circ }

So, the final answer isθ=120\theta = {120^ \circ }(or) θ=2π3\theta = \dfrac{{2\pi }}{3}.

Note: in this type of question we would find the valueθ\theta from the given equation. In this question, we use trigonometric table values to find the final answer. Also, we can use a scientific calculator to find the value . On finding the value in the calculator we can use either radian mode or degree mode. If we want to find the value in the decimal value we can use radian mode. If we want to find the value in the degree we can use degree mode.