Using the diagram below, write the ratio of cos M. 2. What is the complement of cot 30°? 3. Given tan Q = 2/3, find the length of side QR in surd form.

In the given right-angled triangle:

• The side adjacent to angle M is 15.
• The hypotenuse (the side opposite the right angle) is 17.

The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Substitute the values from the diagram:

The ratio of $\cos M$ is $\boxed{\frac{15}{17}}$.

2. What is the complement of $\cot 30^\circ$?

The complement of a trigonometric ratio of an angle is its co-function of the complementary angle. For example, the complement of $\cot \theta$ is $\tan (90^\circ - \theta)$.

The complementary angle of $30^\circ$ is $90^\circ - 30^\circ = 60^\circ$.

The co-function of cotangent is tangent.
Therefore, the complement of $\cot 30^\circ$ is $\tan (90^\circ - 30^\circ) = \tan 60^\circ$.
The complement of $\cot 30^\circ$ is $\boxed{\tan 60^\circ}$.

3. Given $\tan Q = \frac{2}{3}$, find the length of side QR in surd form.

In the given right-angled triangle:

• The side opposite angle Q is labeled 2.
• The side adjacent to angle Q is QR.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Substitute the values from the diagram and the given ratio:

Since the numerators are equal, the denominators must also be equal.

An integer can be expressed in surd form by taking its square root and squaring it.

The length of side QR in surd form is $\boxed{\sqrt{9}}$.