Question

question 11
find the exact value of the expression.
(\tan left(315^{circ}
ight)=)
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question 12
find the exact value of the expression.
(cot left(225^{circ}
ight)=)
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Explanation:

Step1: Find the reference angle for \(315^{\circ}\)

The reference angle \(\theta_{r}=360^{\circ}-315^{\circ} = 45^{\circ}\). Since \(315^{\circ}\) is in the fourth - quadrant, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and in the fourth - quadrant \(\sin\theta<0\), \(\cos\theta>0\), so \(\tan(315^{\circ})=-\tan(45^{\circ})\).

Step2: Calculate \(\tan(45^{\circ})\)

We know that \(\tan(45^{\circ}) = 1\), so \(\tan(315^{\circ})=- 1\).

Step3: Find the reference angle for \(225^{\circ}\)

The reference angle \(\theta_{r}=225^{\circ}-180^{\circ}=45^{\circ}\). Since \(225^{\circ}\) is in the third - quadrant, \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), and in the third - quadrant \(\sin\theta<0\), \(\cos\theta<0\), so \(\cot(225^{\circ})=\cot(45^{\circ})\).

Step4: Calculate \(\cot(45^{\circ})\)

We know that \(\cot(45^{\circ})=\frac{\cos(45^{\circ})}{\sin(45^{\circ})}=1\).

Answer:

\(\tan(315^{\circ})=-1\)
\(\cot(225^{\circ}) = 1\)