Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-18, P\) 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. \(\csc \theta\) c. \(\sec \theta\) d. \(\cot \theta\) \((3,4)\)

Step-by-Step Solution

Answer

a. \(r = 5\) b. \(\csc \theta = \frac{5}{4}\) c. \(\sec \theta = \frac{5}{3}\) d. \(\cot \theta = \frac{3}{4}\)

Step 1: Identify Coordinates

The point \(P\) is given as \((3, 4)\). These coordinates represent the \(x\) and \(y\) values on the circle.

Step 2: Calculate Radius \(r\)

Using the coordinates \((x, y) = (3, 4)\), we calculate \(r\) using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Therefore, the radius \(r\) is 5.

Step 3: Find \(\csc \theta\)

The cosecant is defined as \(\csc \theta = \frac{r}{y}\). Substitute \(r = 5\) and \(y = 4\) to get: \[ \csc \theta = \frac{5}{4}. \]

Step 4: Find \(\sec \theta\)

The secant is defined as \(\sec \theta = \frac{r}{x}\). Using \(r = 5\) and \(x = 3\), we find: \[ \sec \theta = \frac{5}{3}. \]

Step 5: Find \(\cot \theta\)

Cotangent is defined as \(\cot \theta = \frac{x}{y}\). With \(x = 3\) and \(y = 4\), the calculation is: \[ \cot \theta = \frac{3}{4}. \]

Key Concepts

Cosecant

The cosecant function, abbreviated as \( \csc \theta \), is one of the trigonometric functions, and it holds a special relationship with the sine function. In simple terms, it is the reciprocal of the sine function, defined for an angle \( \theta \). This means if you know the sine of an angle, you can easily compute its cosecant and vice versa.

• The formula for calculating cosecant is: \( \csc \theta = \frac{1}{\sin \theta} = \frac{r}{y} \).
• Here, \( r \) is the radius of the circle, which in our solution was 5, and \( y \) is the vertical coordinate, which was 4.
• Using the coordinates we found earlier, we calculated \( \csc \theta = \frac{5}{4} \).

The cosecant function is undefined when \( \theta \) is such that \( \sin \theta = 0 \), because division by zero is undefined in mathematics. This typically happens when \( \theta \) corresponds to angles like 0 degrees or 180 degrees.

Secant

The secant function, written as \( \sec \theta \), shares a similar role with cosine. It is the reciprocal of the cosine function. If you find the cosine of an angle, knowing the secant becomes straightforward.

• The formula for secant is: \( \sec \theta = \frac{1}{\cos \theta} = \frac{r}{x} \).
• From our previous steps, \( r \) was 5 and \( x \) was 3, leading us to calculate \( \sec \theta = \frac{5}{3} \).

Just like cosecant, secant becomes undefined when \( \cos \theta = 0 \). This occurs at angles like 90 degrees and 270 degrees, where the cosine transitions through zero.

Cotangent

Cotangent, denoted as \( \cot \theta \), is analogous to tan and represents the reciprocal relationship. It makes use of the tangent of an angle but flips it to provide a different perspective on trigonometric relationships.

• The mathematical expression for cotangent is: \( \cot \theta = \frac{1}{\tan \theta} = \frac{x}{y} \).
• Here, \( x \) was established as 3 and \( y \) as 4 from our coordinates, providing \( \cot \theta = \frac{3}{4} \).

Keep in mind that cotangent is undefined when the tangent function is zero, or when \( \theta \) is 0 degrees or 180 degrees, since these angles result in a division by zero in the tangent formula.