Problem 19 - 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. \(\tan \theta\) $$ (4,-4) $$

Step-by-Step Solution

Answer

a. \(r = 4\sqrt{2}\); b. \(\cos \theta = \frac{\sqrt{2}}{2}\); c. \(\sin \theta = -\frac{\sqrt{2}}{2}\); d. \(\tan \theta = -1\).

Step 1: Find the Radius \(r\)

The radius \(r\) of the circle can be found using the Pythagorean Theorem. Given the coordinates \((x, y) = (4, -4)\), we calculate: \[ r = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}.\] Thus, the radius \(r\) is \(4\sqrt{2}\).

Step 2: Calculate \(\cos \theta\)

The cosine of the angle \(\theta\) is the ratio of the adjacent side of the triangle to the hypotenuse (radius \(r\)). Here, the adjacent side length is the x-coordinate: \[\cos \theta = \frac{x}{r} = \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.\] Thus, \(\cos \theta\) is \(\frac{\sqrt{2}}{2}\).

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

The sine of the angle \(\theta\) is the ratio of the opposite side to the hypotenuse. Here, the opposite side length is the y-coordinate: \[\sin \theta = \frac{y}{r} = \frac{-4}{4\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}.\] Hence, \(\sin \theta\) is \(-\frac{\sqrt{2}}{2}\).

Step 4: Calculate \(\tan \theta\)

The tangent of the angle \(\theta\) is the ratio of the sine of the angle to the cosine of the angle, which is also the ratio of opposite to adjacent: \[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1.\] Therefore, \(\tan \theta\) is \(-1\).

Key Concepts

Pythagorean Theorem

When working with circles and points in a coordinate plane, the Pythagorean Theorem is super handy. It's like a trusty tool to help find the length of different sides of a right triangle. The theorem states that for any right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In simple terms:

• Hypotenuse = \( r \)
• Other sides = \( x \) and \( y \)

The formula to find the hypotenuse, or radius \( r \), is: \[ r = \sqrt{x^2 + y^2} \]This means you square the two shorter sides and add them together. Then, you take the square root of that sum to find the hypotenuse, \( r \). In our example, the coordinates are \((4, -4)\). Plug those into the equation, and you'll find the radius of the circle. The Pythagorean Theorem is a guiding light for finding distances in a coordinate plane.

cosine

Cosine is one of the basic trigonometric functions, and it's really helpful for linking angles with side ratios in triangles. For a given angle \(\theta\), cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle:

• Adjacent side = \( x \)
• Hypotenuse = radius \( r \)

The formula to find \(\cos \theta\) is:\[\cos \theta = \frac{x}{r}\]Using our coordinates \((4, -4)\), we set \(x = 4\) and our radius \(r = 4\sqrt{2}\). Plug these into the equation:\[\cos \theta = \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]So, the cosine of the angle \(\theta\) is \(\frac{\sqrt{2}}{2}\). The cosine function helps us understand how steep a slant in a coordinate plane is.

sine

Sine is another core trigonometric function that connects angles with side ratios in triangles. It represents the ratio of the opposite side to the hypotenuse for a given angle \( \theta \):

• Opposite side = \( y \)
• Hypotenuse = radius \( r \)

To find \(\sin \theta\), use:\[\sin \theta = \frac{y}{r}\]For coordinates \((4, -4)\), \(y = -4\) and the radius \(r = 4\sqrt{2}\). The calculation becomes:\[\sin \theta = \frac{-4}{4\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}\]This tells us that \(\sin \theta\) is \(-\frac{\sqrt{2}}{2}\). Sine is a great way to measure vertical distances in a unit circle context or in triangles.

tangent

Tangent is a very interesting function in trigonometry that relates to both sine and cosine. It tells you about the steepness or incline of a line formed by \(\theta\). Mathematically, tangent for an angle \(\theta\) is:

• It represents the ratio of sine to cosine.
• It's also the ratio of the opposite side to the adjacent side.

To express this formula:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}\]Using our values, where \(y = -4\), \(x = 4\), \(\sin \theta = -\frac{\sqrt{2}}{2}\), and \(\cos \theta = \frac{\sqrt{2}}{2}\), we find:\[\tan \theta = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1\]This shows that the tangent of \(\theta\) is \(-1\). Tangent helps us to understand the rate of change or the slope for angles, especially in right triangles and circles.

radius of a circle

The radius of a circle is a crucial element. It's the distance from the center to any point on the circle. In trigonometry, especially when dealing with angles in standard position, understanding the radius helps compute various trigonometric functions.- The radius \( r \) forms the hypotenuse in a right triangle when looking at a point on a circle.- It is deeply tied to the Pythagorean Theorem, which is used to find \( r \) by calculating the distance using coordinates.For example, if you have a point \((x, y)\), the radius \( r \) is:\[r = \sqrt{x^2 + y^2}\]In our exercise with the point \((4, -4)\), following this formula, we find that:\[r = \sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}\]The radius not only defines the size of the circle but also ties the geometric position with trigonometric functions, making it essential in solving such exercises effectively.