Problem 43 - Algebra 2

Question

Suppose \(\theta\) is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of \(\theta .\) \(\tan \theta=-\frac{1}{5},\) Quadrant \(\Pi\)

Step-by-Step Solution

Answer

\( \sin \theta = \frac{\sqrt{26}}{26}, \cos \theta = -\frac{5\sqrt{26}}{26}, \csc \theta = \sqrt{26}, \sec \theta = -\frac{\sqrt{26}}{5}, \cot \theta = -5. \)

Step 1: Understanding Quadrant II

In Quadrant II, the sine function, which represents the vertical leg in the unit circle, is positive. Cosine and tangent are negative. This knowledge will help us determine the signs of each trigonometric ratio.

Step 2: Using the given tangent value

We are given \( \tan \theta = -\frac{1}{5} \). Tangent is the ratio of the sine and cosine, i.e., \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). In terms of the sides of a right triangle, this implies that for some angle \( \theta \), the opposite side is 1 (or -1 by the direction) and the adjacent side is -5.

Step 3: Applying the Pythagorean Identity

Using the Pythagorean Theorem, we can find the hypotenuse. The equation is \( a^2 + b^2 = c^2 \). For the triangle sides, \( a = 1 \) and \( b = -5 \), we have: \( 1^2 + (-5)^2 = c^2 \), simplifying to \( 1 + 25 = c^2 \), thus \( c^2 = 26 \) and \( c = \sqrt{26} \).

Step 4: Finding exact values for sine and cosine

Since \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), we have \( \sin \theta = \frac{1}{\sqrt{26}} \). Enabling exact representation, it is \( \frac{\sqrt{26}}{26} \) after rationalization.For cosine, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), it follows \( \cos \theta = \frac{-5}{\sqrt{26}} = -\frac{5\sqrt{26}}{26} \) after rationalization.

Step 5: Calculating the remaining trigonometric functions

The cosecant is the reciprocal of sine: \( \csc \theta = \frac{\sqrt{26}}{1} = \sqrt{26} \). The secant is the reciprocal of cosine: \( \sec \theta = -\frac{\sqrt{26}}{5} \).The cotangent, which is the reciprocal of tangent, is \( \cot \theta = -5 \).

Key Concepts

Tangent

Tangent, often abbreviated as tan, is one of the primary trigonometric functions and is crucial in understanding angles and triangles. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side. This gives us the formula:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

In the context of the unit circle, which we will discuss later, tangent can be represented as the y-coordinate divided by the x-coordinate for a point on the circle.
Tangent's sign depends on the quadrant in which the angle's terminal side lies. In the second quadrant (Quadrant II), tangent is negative because the sine is positive and cosine is negative. Remember, tangent combines these two factors:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Understanding the properties of tangent is essential for solving various trigonometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry and geometry. It applies to right-angled triangles and establishes the relationship between the sides of the triangle. The formula is straightforward:

• \( a^2 + b^2 = c^2 \)

Here, \(a\) and \(b\) are the lengths of the two legs of the triangle, and \(c\) is the hypotenuse.
In trigonometry, this theorem can be used to find any side of a right triangle if you know the lengths of the other two sides.
In the given problem, with tangent, we deduced the lengths of the opposite and adjacent sides and used the Pythagorean Theorem to compute the hypotenuse as \( \sqrt{26} \).
This calculation is crucial as it allows us to find the other trigonometric functions such as sine and cosine, opening doors to further investigations with the trigonometric identities.

Unit Circle

The unit circle is a vital concept in trigonometry, helping visualize and understand the values of trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
Each point on the unit circle corresponds to an angle \(\theta\) from the positive x-axis, and the coordinates of these points are \((\cos \theta, \sin \theta)\).
This simple relationship makes the unit circle an excellent tool for understanding all trigonometric functions, especially sine and cosine.

• The x-coordinate gives \(\cos \theta\).
• The y-coordinate gives \(\sin \theta\).

For tangent, which is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the unit circle helps easily determine its value and sign. The angle's position on the unit circle quickly tells you which functions are positive or negative based on the quadrant.

Reciprocal Functions

Reciprocal functions are an extension of the basic trigonometric functions and are very helpful in various calculations. There are three main reciprocal functions: cosecant, secant, and cotangent.

• Cosecant, \( \csc \theta \), is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant, \( \sec \theta \), is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent, \( \cot \theta \), is the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \).

These functions inherit the sign properties from their respective base functions. For instance, in our problem where \( \tan \theta = -\frac{1}{5} \), the \( \cot \theta \) will be \(-5\), reflecting the reciprocal relationship within the constraints of the second quadrant where tangent is negative.