Problem 13 - 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. \(\cot \theta\) \((-3,-3)\)

Step-by-Step Solution

Answer

\( r = 3\sqrt{2}, \csc \theta = -\sqrt{2}, \sec \theta = -\sqrt{2}, \cot \theta = 1 \).

Step 1: Calculate the Radius, r

To find the radius \( r \) of the circle, we use the distance formula from the origin \( (0,0) \) to the point \( (-3, -3) \). This is given by \( r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \).

Step 2: Determine the Sine, Cosine, and Tangent Values

The coordinates of the point \((-3, -3)\) give us \( x = -3 \) and \( y = -3 \). The sine and cosine of \( \theta \) are found using \( \sin \theta = \frac{y}{r} = \frac{-3}{3\sqrt{2}} = -\frac{\sqrt{2}}{2} \) and \( \cos \theta = \frac{x}{r} = \frac{-3}{3\sqrt{2}} = -\frac{\sqrt{2}}{2} \). The tangent is \( \tan \theta = \frac{y}{x} = \frac{-3}{-3} = 1 \).

Step 3: Calculate the Cosecant, \( \csc \theta \)

The cosecant function is the reciprocal of sine. Thus, \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \).

Step 4: Calculate the Secant, \( \sec \theta \)

The secant function is the reciprocal of cosine. Thus, \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \).

Step 5: Calculate the Cotangent, \( \cot \theta \)

The cotangent function is the reciprocal of tangent. Thus, \( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{1} = 1 \).

Key Concepts

Cosecant

The cosecant, abbreviated as \( \csc \theta \), is one of the three important reciprocal trigonometric functions. It's the reciprocal of the sine function. In simpler terms, if you know the sine of an angle, you can find its cosecant by flipping the fraction upside down.
### Practical Understanding of CosecantIf \( \sin \theta = \frac{y}{r} \), then \( \csc \theta = \frac{r}{y} \). For example, if \( \sin \theta = -\frac{\sqrt{2}}{2} \) as per the exercise, the cosecant will be:

• Flip the sine: \( \csc \theta = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \)

Why is this useful? The cosecant, along with other trigonometric functions, helps in solving various geometry and calculus problems. You can always visualize it as the "height vs. distance" relationship on the radius of a circle.
Knowing \( \csc \theta \) can help when calculating lengths and angles in right-angled triangles, especially when other direct trigonometric values are not readily available.

Secant

The secant function, denoted as \( \sec \theta \), is another reciprocal trigonometric function. It is the reciprocal of the cosine function. Just like the cosecant and sine relationship, secant is about reversing the cosine value.
### Practical Understanding of SecantFor \( \cos \theta = \frac{x}{r} \), the secant can be found as \( \sec \theta = \frac{r}{x} \). In our specific case from the exercise, since \( \cos \theta = -\frac{\sqrt{2}}{2} \), the calculation is:

• Flip the cosine: \( \sec \theta = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \)

Secant is particularly useful in scenarios involving stretching or compressing of dimensions in geometric problems.
It is often visualized on the unit circle where you see the length of the line segment from the origin to a point where the circle intersects a line parallel to the y-axis. Thus, it plays a critical role in understanding scenarios where angles and distances are directly related among circles and triangles.

Cotangent

Cotangent, abbreviated as \( \cot \theta \), is the reciprocal of the tangent function. This means it flips the roles of sine and cosine compared to tangent. Tangent is normally \( \tan \theta = \frac{y}{x} \), while the cotangent changes this to \( \cot \theta = \frac{x}{y} \).
### Solving for CotangentTaking the exercise example, where \( \tan \theta = 1 \), cotangent is straightforward:

• Flip the tangent: \( \cot \theta = \frac{1}{1} = 1 \)

Cotangent has great use in trigonometry, especially when dealing with angles related to circles and periodic functions.
It's often helpful in determining complementary angles and understanding the transition between different trigonometric ratios in various quadrants of the trigonometric circle.Understanding cotangent can greatly ease calculations, particularly in problems where angles are approaching 0 or \( \pi \), due to its relationships and properties with broader trigonometric identities.