Question

1 multiple choice 1 point given the triangle xyz, what is cos z? 2 multiple choice 1 point find a.

Explanation:

Step1: Recall the cosine formula

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For angle $Z$ in right - triangle $XYZ$, the adjacent side to angle $Z$ is $YZ = 30$ and the hypotenuse is $XZ=34$.

Step2: Calculate $\cos Z$

$\cos Z=\frac{30}{34}=\frac{15}{17}$

Step1: Identify the type of right - triangle

Since one of the non - right angles is $45^{\circ}$, the right - triangle is a $45 - 45-90$ triangle. In a $45 - 45 - 90$ triangle, the legs are equal. Let the legs be $a$ and $b$ and the hypotenuse be $c$. The relationship is $a = b$ and $c=\sqrt{2}a$. Here, if we assume the given side (let's say adjacent to the $45^{\circ}$ angle) is one of the legs. But wait, using trigonometry: $\sin45^{\circ}=\frac{a}{c}$ or $\cos45^{\circ}=\frac{26}{c}$ (where $c$ is the hypotenuse). Also, another way: since it's a right - triangle, using the Pythagorean theorem $a^{2}+26^{2}=c^{2}$, and for a $45 - 45 - 90$ triangle $c=\sqrt{2}a$. Substituting $c = \sqrt{2}a$ into $a^{2}+26^{2}=c^{2}$ gives $a^{2}+676 = 2a^{2}$, then $a^{2}=676$, $a = 26$.

Answer:

$\frac{15}{17}$ (corresponding to the third option)

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