Question

4 multiple choice 1 point give the exact value. cos 45° 5 multiple choice 1 point give the exact value. sec 150°

Explanation:

Step1: Recall the cosine value of special angles

We know that for a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, if the legs are of length \(a\) and the hypotenuse is \(c\), by the Pythagorean theorem \(c^{2}=a^{2}+a^{2}=2a^{2}\), so \(c = a\sqrt{2}\). Then \(\cos45^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{a}{a\sqrt{2}}=\frac{1}{\sqrt{2}}\). Rationalizing the denominator, we get \(\frac{\sqrt{2}}{2}\).

Step2: Recall the secant formula and reference angle for \(150^{\circ}\)

We know that \(\sec\theta=\frac{1}{\cos\theta}\). The angle \(150^{\circ}\) has a reference angle of \(180 - 150=30^{\circ}\). And \(\cos150^{\circ}=-\cos30^{\circ}\) (since \(150^{\circ}\) is in the second - quadrant where cosine is negative). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), then \(\cos150^{\circ}=-\frac{\sqrt{3}}{2}\). So \(\sec150^{\circ}=\frac{1}{\cos150^{\circ}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}\). Rationalizing the denominator gives \(-\frac{2\sqrt{3}}{3}\).

Answer:

For \(\cos45^{\circ}\): \(\frac{\sqrt{2}}{2}\) (the fourth option).
For \(\sec150^{\circ}\): \(-\frac{2\sqrt{3}}{3}\) (the second option).