Problem 1 - Algebra and Trigonometry

Question

If the angle \(\theta\) is in standard position and \(P(x, y)\) is a point on the terminal side of \(\theta,\) and \(r\) is the distance from the origin to \(P,\) then \(\sin \theta=\) \(\frac{\square}{\square}\) \(\cos \theta=\) \(\frac{\square}{\square}\) \(\tan \theta=\) \(\frac{\square}{\square}\)

Step-by-Step Solution

Answer

\(\sin \theta = \frac{y}{r}\), \(\cos \theta = \frac{x}{r}\), \(\tan \theta = \frac{y}{x}\).

Step 1: Understand the problem

We need to find the trigonometric functions \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) for an angle \(\theta\) in standard position, with point \(P(x, y)\) on its terminal side and \(r\) the distance from the origin to \(P\).

Step 2: Determine the distance r from the origin to P(x, y)

The distance \(r\) is calculated using the distance formula from the origin to the point \(P(x, y)\). It is given by \( r = \sqrt{x^2 + y^2} \).

Step 3: Define \(\sin \theta\)

The sine of the angle \(\theta\), \(\sin \theta\), is defined as the ratio of the opposite side to the hypotenuse in the context of a right triangle, formed by the x-axis and line OP. Here it is \(\sin \theta = \frac{y}{r}\).

Step 4: Define \(\cos \theta\)

The cosine of the angle \(\theta\), \(\cos \theta\), is the ratio of the adjacent side to the hypotenuse. Thus, \(\cos \theta = \frac{x}{r}\).

Step 5: Define \(\tan \theta\)

The tangent of the angle \(\theta\), \(\tan \theta\), is the ratio of the opposite side to the adjacent side. Therefore, \(\tan \theta = \frac{y}{x}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding the relationships between the angles and sides of a triangle. These functions are particularly useful when dealing with angles in a standard position. A standard position angle is one whose vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The trigonometric functions we commonly deal with include sine (\(\sin(\theta)\)), cosine (\(\cos(\theta)\)), and tangent (\(\tan(\theta)\)). Each of these functions is based on the properties of a right triangle formed by the terminal side of the angle and the coordinate axes.

• Sine is defined as the ratio of the length of the opposite side to the hypotenuse.
• Cosine is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent is the ratio of the length of the opposite side to the adjacent side.

Understanding these functions allows us to solve many problems in physics, engineering, and other fields. They can also be used to describe wave patterns, circular motion, and many other phenomena in nature.

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry and have specific definitions based on their geometrical interpretations. Given a point \(P(x, y)\) on the terminal side of an angle \(\theta\), the distance from the origin to this point is denoted as \(r\), which acts as the hypotenuse of our right triangle.

To compute the sine of angle \(\theta\):

• The angle has a vertical side (opposite) of length \(y\),
• and a hypotenuse of length \(r\),
• so \(\sin(\theta) = \frac{y}{r}\).

For the cosine of angle \(\theta\):

• The horizontal side (adjacent) has a length of \(x\),
• with the same hypotenuse of length \(r\),
• meaning \(\cos(\theta) = \frac{x}{r}\).

These ratios \(\sin(\theta)\) and \(\cos(\theta)\) are crucial in determining the properties of rotational motion and in analyzing models that involve circular paths or oscillations, such as sound waves or light waves.

Distance Formula

The distance formula is a method used to determine the length of a line segment between two points in a coordinate plane. For any points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance \(d\) between them is calculated as:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
In the context of trigonometry and angles in standard position, we use this formula to find the distance \(r\) from the origin \((0, 0)\) to the point \(P(x, y)\) on the terminal side of \(\theta\). Therefore, \(r\) is the hypotenuse of the right triangle,
\[r = \sqrt{x^2 + y^2}\].
This step is crucial as it allows us to use the hypotenuse, \(r\), in calculating the sine and cosine of the angle \(\theta\). By knowing \(r\), we can convert the Cartesian coordinates \((x, y)\) into polar form, facilitating various forms of mathematical calculations and analyses such as in polar graphs or physics scenarios involving radial vectors.