91Ó°ÊÓ

The Pythagorean theorem is a fundamental concept in trigonometry and geometry. It is crucial when determining the distance of a point from the origin on a coordinate plane. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This can be expressed with the equation:

• \(a^2 + b^2 = c^2\)

where:

• \(a\) and \(b\) are the lengths of the triangle's legs,
• and \(c\) is the length of the hypotenuse.

In our problem, the coordinates (1, \(\sqrt{15}\)) are used to form a right triangle where \(x = 1\) and \(y = \sqrt{15}\). Using the Pythagorean theorem:

• \[r = \sqrt{1^2 + (\sqrt{15})^2} = \sqrt{16} = 4\].

This resulting \(r\) represents the hypotenuse or the distance of the point from the origin. The hypotenuse is essential for calculating other trigonometric function values.

The sine function is a fundamental trigonometric function, representing the ratio of the length of the side opposite to the angle to the hypotenuse in a right triangle. For an angle \(\theta\) in standard position, with a terminal side passing through the point \((1, \sqrt{15})\), the sine can be calculated using the following expression:

• \(\sin(\theta) = \frac{y}{r}\)

Here, \(y = \sqrt{15}\) and \(r = 4\). Thus:

• \[\sin(\theta) = \frac{\sqrt{15}}{4}\].

This ratio helps in understanding how the angle \(\theta\) relates to the height of a vertical line drawn from the x-axis to the point \((1, \sqrt{15})\). The sine is crucial in various real-world applications such as waves, vibrations, and circular motion.

The cosine function is another key trigonometric function that measures the ratio of the adjacent side to the hypotenuse in a right triangle. When positioning an angle in the standard position, you can compute the cosine using:

• \( \cos(\theta) = \frac{x}{r} \)

For our example, with \(x = 1\) and \(r = 4\), the formula gives:

• \[ \cos(\theta) = \frac{1}{4} \].

This measure provides insight into the horizontal distance from the origin to the position where the terminal side meets the circle with radius \(r\). Understanding the cosine is vital for contexts where the length along the ground is a factor, such as in physics and engineering problems involving force and motion.

The tangent function links the sine and cosine functions by expressing their ratio. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. For our angle \(\theta\) in standard position, tangent is computed by:

• \( \tan(\theta) = \frac{y}{x} \)

With \(y = \sqrt{15}\) and \(x = 1\), this gives:

• \[ \tan(\theta) = \frac{\sqrt{15}}{1} = \sqrt{15} \].

The tangent function provides a way to calculate the "steepness" or "slope" of the line connecting the origin to the point \((1, \sqrt{15})\). It is widely used in calculations involving angles and slopes, such as in navigation, architecture, and even computer graphics.