In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

In ΔPQR, we are given:

• ∠R = 90° (so it's a right triangle),
• PR = 56 (this is one of the legs),
• RQ = 33 (this is the other leg),
• QP = 65 (this is the hypotenuse, opposite ∠R).

To find the cosine of ∠P, we use the definition of cosine: $\cos(\angle P) = \frac{\text{adjacent}}{\text{hypotenuse}}.$ Here, the adjacent side to ∠P is RQ (33), and the hypotenuse is QP (65). Therefore: $\cos(\angle P) = \frac{RQ}{QP} = \frac{33}{65}.$

Thus, the ratio representing the cosine of ∠P is $\frac{33}{65}$.

Would you like further clarification or have any questions on this? Here are some related questions:

1. How do you calculate the sine of an angle in a right triangle?
2. What is the Pythagorean theorem, and how is it used in this problem?
3. How can you verify if the sides of a triangle form a right triangle using the Pythagorean theorem?
4. What other trigonometric ratios can be used to find angles in right triangles?
5. How do you calculate the tangent of ∠P?

Tip: To easily remember trigonometric ratios in a right triangle, you can use the mnemonic SOH CAH TOA:

• Sine = Opposite / Hypotenuse,
• Cosine = Adjacent / Hypotenuse,
• Tangent = Opposite / Adjacent.