Use the triangle to find each of the six trigonometric functions of θ.

Verified step by step guidance

1. Identify the sides of the right triangle: MN = 11 (adjacent), ON = 4 (opposite), and MO (hypotenuse) is unknown.
2. Use the Pythagorean theorem to find the hypotenuse: MO = \( \sqrt{MN^2 + ON^2} \).
3. Calculate the sine of \( \alpha \): \( \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{ON}{MO} \).
4. Calculate the cosine of \( \alpha \): \( \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{MN}{MO} \).
5. Calculate the tangent of \( \alpha \): \( \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{ON}{MN} \).

Key Concepts

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are fundamental in relating the angles and sides of a triangle. For a right triangle, these functions are defined as ratios of the lengths of the sides. For example, sine(θ) is the ratio of the opposite side to the hypotenuse, while cosine(θ) is the ratio of the adjacent side to the hypotenuse.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for finding the length of the hypotenuse when the lengths of the other two sides are known, which is crucial for calculating the trigonometric functions.

Angle of Elevation

The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line to an object above. In the context of the triangle provided, angle α is the angle of elevation from point N to point O. Understanding this angle is important for applying trigonometric functions to solve for unknown sides or angles in the triangle.