Problem 20 - Algebra 2 and Trigonometry

Question

In \(13-20, P\) is a point on the terminal side of an angle in standard position with measure \(\theta\) and on a circle with center at the origin and radius \(r .\) For each point \(P,\) find: a. \(r\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) $$ (9,-3) $$

Step-by-Step Solution

Answer

a. \(r = 3\sqrt{10}\), b. \(\cos \theta = \frac{\sqrt{10}}{3}\), c. \(\sin \theta = \frac{-3}{10}\), d. \(\tan \theta = -1/3\)

Step 1: Understanding the given point

The point given is \((9, -3)\), which lies on the terminal side of an angle \(\theta\) in standard position. The point is on the circle centered at the origin \((0,0)\) with radius \(r\).

Step 2: Calculating the radius \(r\)

The radius \(r\) is the distance from the origin to the point \((9, -3)\). Use the distance formula: \[ r = \sqrt{(x^2 + y^2)} = \sqrt{(9^2 + (-3)^2)} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10} \]

Step 3: Determining \(\cos \theta\)

The cosine of angle \(\theta\) is the x-coordinate divided by the radius: \[ \cos \theta = \frac{x}{r} = \frac{9}{3\sqrt{10}} = \frac{3\sqrt{10}}{10} \] Rationalizing the denominator, we get: \[ \cos \theta = \frac{3\sqrt{10} \times \sqrt{10}}{10 \times \sqrt{10}} = \frac{3\sqrt{10}}{30} = \frac{\sqrt{10}}{3} \]

Step 4: Determining \(\sin \theta\)

The sine of angle \(\theta\) is the y-coordinate divided by the radius: \[ \sin \theta = \frac{y}{r} = \frac{-3}{3\sqrt{10}} = \frac{-\sqrt{10}}{10} \] Rationalizing the denominator, we get: \[ \sin \theta = \frac{-\sqrt{10} \times \sqrt{10}}{10 \times \sqrt{10}} = \frac{-3}{10} \]

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

The tangent of angle \(\theta\) is the sine divided by cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3}{10}\cdot\frac{3}{\sqrt{10}} = \left(\frac{-\sqrt{10}}{3}\right)\cdot\left(\frac{3}{\sqrt{10}}\right) = \frac{-1}{3} \]

Key Concepts

Standard position

In trigonometry, an angle is said to be in "standard position" when its vertex is located at the origin \((0, 0)\) of a coordinate plane, and its initial side lies along the positive x-axis.The terminal side of the angle is the ray where the angle ends.For example, if a point \(P(x, y)\) like \((9, -3)\) lies on the terminal side of an angle, it means the angle is formed between the positive x-axis and the line connecting \(P\) to the origin.Understanding standard position is essential because:

• It provides a consistent reference point for measuring angles.
• It helps in applying trigonometric ratios precisely.
• Makes it easier to visualize and solve problems related to angles, such as those involving \(x\) and \(y\) coordinates.

Determining how points fit in this system forms the basis for calculating trigonometric functions like cosine, sine, and tangent.

Cosine

The cosine (\(\cos\)) of an angle is one of the fundamental trigonometric functions. In the context of angles in standard position, \(\cos \theta\) is defined as the ratio of the adjacent side (x-coordinate) to the hypotenuse (radius).If you imagine a right triangle that shares the x-axis as one of its sides, the hypotenuse is the line from the origin to the point \(P(x, y)\).For example:

• In the case where \(P = (9, -3)\), the cosine of the angle \(\theta\) can be calculated using: \[\cos \theta = \frac{x}{r} = \frac{9}{3\sqrt{10}} = \frac{3\sqrt{10}}{10}.\]
• Rationalizing results in \(\cos \theta = \frac{\sqrt{10}}{3}.\)

Knowing \(\cos \theta\) is useful for understanding how far along the x-axis your terminal side reaches as it helps in understanding the angle's behavior on the x-axis.

Sine

The sine (\(\sin\)) function is another core trigonometric ratio, and is used to express the length of the side opposite the angle divided by the hypotenuse.In standard position, \(\sin \theta\) is represented as the y-coordinate over the radius that reaches the point \(P\).Using our example point \((9, -3)\), we find:

• \[\sin \theta = \frac{y}{r} = \frac{-3}{3\sqrt{10}} = \frac{-\sqrt{10}}{10}.\]
• Rationalizing this gives \(\sin \theta = \frac{-3}{10}.\)

The sine provides insight into vertical displacement from the x-axis, determining how the angle stretches on the vertical plane.

Tangent

The tangent (\(\tan\)) function blends both the sine and cosine functions into a meaningful ratio.It is essentially the ratio of the opposite side to the adjacent side of the reference triangle.When examining angles in standard position:\[\tan \theta = \frac{\sin \theta}{\cos \theta}.\]Taking our example point \((9, -3)\):

• \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3/10}{\sqrt{10}/3} = -1/3.\)

The tangent ratio can tell you about the slope or steepness of the line that represents the angle in standard position.A positive \(\tan \theta\) implies the line rises from left to right, while a negative value as here means it falls.