Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator
225°
a.
sin 225° = -\frac{\sqrt{2}}{2}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )
b. the function value is undefined
select the correct choice below and fill in any answer boxes within your choice
a.
cos 225° = -\frac{\sqrt{2}}{2}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )
b. the function value is undefined
select the correct choice below and fill in any answer boxes within your choice
a. tan 225° =
(simplify your answer, including any radicals use integers or fractions for any numbers in the expression.)
b. the function value is undefined

Explanation:

Step1: Determine the reference angle

The reference angle for \(225^{\circ}\) is \(225^{\circ}- 180^{\circ}=45^{\circ}\).

Step2: Recall the sign of tangent in the third quadrant

In the third quadrant (\(180^{\circ}<\theta <270^{\circ}\)), tangent is positive (\(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and both \(\sin\theta\) and \(\cos\theta\) are negative in the third quadrant, so their ratio is positive).

Step3: Calculate the tangent value

Since \(\tan45^{\circ} = 1\), and \(\tan225^{\circ}=\tan(180^{\circ} + 45^{\circ})\), and using the identity \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\) with \(A = 180^{\circ}\) (\(\tan180^{\circ}=0\)) and \(B = 45^{\circ}\), we have \(\tan225^{\circ}=\frac{0 + 1}{1-0\times1}=1\). Also, using the ratio \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), with \(\sin225^{\circ}=-\frac{\sqrt{2}}{2}\) and \(\cos225^{\circ}=-\frac{\sqrt{2}}{2}\), \(\tan225^{\circ}=\frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1\).

Answer:

\(\tan225^{\circ}=1\)