Question

evaluate without using a calculator by using ratios in a reference triangle. tan\\(\frac{3\pi}{4}\\) give the exact value. cos\\(\frac{\pi}{4}\\)

Explanation:

Step1: Analyze \(\tan\frac{3\pi}{4}\)

The angle \(\frac{3\pi}{4}\) is in the second - quadrant. The reference angle \(\theta'=\pi-\frac{3\pi}{4}=\frac{\pi}{4}\). In the second - quadrant, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and \(\sin\theta>0\), \(\cos\theta < 0\). We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = \frac{\pi}{4}\), \(\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\). So, \(\tan\frac{3\pi}{4}=\frac{\sin\frac{3\pi}{4}}{\cos\frac{3\pi}{4}}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=-1\)

Step2: Analyze \(\cos\frac{\pi}{4}\)

For a right - isosceles triangle (a \(45 - 45-90\) triangle) with legs of length \(a\) and hypotenuse \(c\), by the Pythagorean theorem \(c=\sqrt{a^{2}+a^{2}}=\sqrt{2a^{2}}=\sqrt{2}a\). The cosine of an angle in a right triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta=\frac{\pi}{4}\) (or \(45^{\circ}\)), \(\cos\frac{\pi}{4}=\frac{a}{\sqrt{2}a}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\) (rationalizing the denominator)

Answer:

For \(\tan\frac{3\pi}{4}\):

• D. \(- 1\)

For \(\cos\frac{\pi}{4}\):

• \(\frac{\sqrt{2}}{2}\) (Since the options for \(\cos\frac{\pi}{4}\) are not fully visible, but the exact value is \(\frac{\sqrt{2}}{2}\))