Question

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

Answer 1

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

Complete step by step solution:

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

The above figure represents a triangle marked with the opposite side, adjacent side, and hypotenuse
side according to the position of\[\theta \].
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 \[\left( 1 \right)\]and\[\left( 3 \right)\], we get the value of the adjacent
side is\[ - 1\] and the value of the hypotenuse side is\[2\].
When the term \[\cos \]is the move from the left side to the right side of the equation it converts
into\[\arccos \]. Let’s find the value of \[\theta \]from the equation\[\left( 3 \right)\], we get
\[\left( 3 \right) \to \cos \theta = \dfrac{{ - 1}}{2}\]
\[\theta = \arccos \left( {\dfrac{{ - 1}}{2}} \right)\]
By, using the trigonometric table value, we get
\[\cos \left( {{{60}^ \circ }} \right) = \dfrac{1}{2}\]
\[cos\left( {{{120}^ \circ }} \right) = \dfrac{{ - 1}}{2}\]
So we get,
\[\arccos \left( {\dfrac{{ - 1}}{2}} \right) = {120^ \circ }\]\[ \to \left( 4 \right)\]

So, the value of\[\theta \] becomes,
\[\theta= arccos(-0.5)0.5\]

\[\theta = {120^ \circ }\]

So, the final answer is\[\theta = {120^ \circ }\](or) \[\theta = \dfrac{{2\pi }}{3}\].

Note: in this type of question we would find the value\[\theta \]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.