Problem 9 - Contemporary Precalculus

Question

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(-5,12)$$

Step-by-Step Solution

Answer

Question: Given a point (-5, 12) on the terminal side of an angle in standard position, find the six trigonometric functions at angle t. Answer: sin(t) = \(\frac{12}{13}\), cos(t) = \(\frac{-5}{13}\), tan(t) = \(-\frac{12}{5}\), csc(t) = \(\frac{13}{12}\), sec(t) = \(-\frac{13}{5}\), cot(t) = \(-\frac{5}{12}\).

Step 1: Calculate the radius (r) using the Pythagorean theorem

Given the point (-5, 12) has coordinates (x, y), we can calculate the radius using the Pythagorean theorem, which states that \(r^2 = x^2 + y^2\). Plug in x = -5 and y = 12. $$r^2=(-5)^2+(12)^2$$ $$r^2=25+144$$ $$r^2=169$$ $$r=13$$ So, the radius (r) is 13.

Step 2: Find sine function (sin t)

To find the sine function, we use the formula \(sin(t)=\frac{y}{r}\), where y is the y-coordinate (12) and r is the radius (13). $$sin(t)=\frac{12}{13}$$

Step 3: Find cosine function (cos t)

To find the cosine function, we use the formula \(cos(t)=\frac{x}{r}\), where x is the x-coordinate (-5) and r is the radius (13). $$cos(t)=\frac{-5}{13}$$

Step 4: Find tangent function (tan t)

To find the tangent function, we use the formula \(tan(t)=\frac{y}{x}\) or \(tan(t)=\frac{sin(t)}{cos(t)}\). We already found sin(t) and cos(t) in step 2 and 3. $$tan(t)=\frac{\frac{12}{13}}{\frac{-5}{13}}$$ $$tan(t)=-\frac{12}{5}$$

Step 5: Find cosecant function (csc t)

The cosecant function is the reciprocal of the sine function. We found sin(t) in step 2. $$csc(t)=\frac{1}{sin(t)}$$ $$csc(t)=\frac{1}{\frac{12}{13}}$$ $$csc(t)=\frac{13}{12}$$

Step 6: Find secant function (sec t)

The secant function is the reciprocal of the cosine function. We found cos(t) in step 3. $$sec(t)=\frac{1}{cos(t)}$$ $$sec(t)=\frac{1}{\frac{-5}{13}}$$ $$sec(t)=-\frac{13}{5}$$

Step 7: Find cotangent function (cot t)

The cotangent function is the reciprocal of the tangent function. We found tan(t) in step 4. $$cot(t)=\frac{1}{tan(t)}$$ $$cot(t)=\frac{1}{-\frac{12}{5}}$$ $$cot(t)=-\frac{5}{12}$$ Thus, sin(t) = \(\frac{12}{13}\), cos(t) = \(\frac{-5}{13}\), tan(t) = \(-\frac{12}{5}\), csc(t) = \(\frac{13}{12}\), sec(t) = \(-\frac{13}{5}\), and cot(t) = \(-\frac{5}{12}\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry, useful for finding the hypotenuse of a right-angled triangle. The theorem states that in a right triangle, the square of the length of the hypotenuse \( (r) \) is equal to the sum of the squares of the lengths of the other two sides (x and y). Mathematically, this is represented as:
\[ r^2 = x^2 + y^2 \]In the exercise, we use this theorem to calculate the radius \( r \). With the coordinates of the point given as \((-5, 12)\), we plug in:

• \(x = -5\)
• \(y = 12\)

This gives us:
\[ r^2 = (-5)^2 + (12)^2 = 25 + 144 = 169 \]Thus, \( r = \sqrt{169} = 13 \).The radius \( r \) is essential for determining the trigonometric functions of the angle.

Sine Function

The sine function represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In terms of a unit circle, it's the y-coordinate of the corresponding point.For an angle \( t \) in standard position with reference to the origin, the sine of \( t \) is:
\[ \sin(t) = \frac{y}{r} \]where:

• \( y \) is the y-coordinate of the given point.
• \( r \) is the hypotenuse, or radius, calculated using the Pythagorean theorem.

In our exercise, \( y = 12 \) and \( r = 13 \). Therefore, the sine of the angle \( t \) is:
\[ \sin(t) = \frac{12}{13} \]Sine values range from -1 to 1, indicating how far vertically the point is from the origin on the unit circle.

Cosine Function

The cosine function conveys the ratio of the length of the adjacent side to the hypotenuse in a triangle. If visualizing on a unit circle, it represents the x-coordinate.Mathematically, for an angle \( t \), the cosine function is defined as:
\[ \cos(t) = \frac{x}{r} \]where:

• \( x \) is the x-coordinate of the given point.
• \( r \) is the hypotenuse, found using the Pythagorean theorem.

In the example exercise:

• \( x = -5 \)
• \( r = 13 \)

Substituting these values gives:
\[ \cos(t) = \frac{-5}{13} \]Cosine values also range from -1 to 1 and determine the horizontal distance of the point from the origin.

Tangent Function

The tangent function is the ratio of the sine function to the cosine function, equivalent to the opposite side to the adjacent side of a triangle.In simpler words, for a given angle \( t \), tangent is calculated as:
\[ \tan(t) = \frac{y}{x} \]Alternatively, using trigonometric identities:
\[ \tan(t) = \frac{\sin(t)}{\cos(t)} \]This relationship highlights tangent's connection to both sine and cosine, combining them into a single expression.In the provided solution:

• \( y = 12 \)
• \( x = -5 \)

This results in:
\[ \tan(t) = \frac{12}{-5} = -\frac{12}{5} \]Since tangent can take any real value, it indicates the slope of the line drawn from the origin to the point \((x, y)\) on the unit circle.