Problem 29 - Precalculus Essentials

Question

Find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\tan \theta=-\frac{2}{7}, \quad \sin \theta>0$$

Step-by-Step Solution

Answer

The exact values of the remaining trigonometric functions are: \( \sin \theta=\frac{-2}{\sqrt{53}}, \cos \theta=\frac{7}{\sqrt{53}}, \csc \theta=-\frac{\sqrt{53}}{2}, \sec \theta=\frac{\sqrt{53}}{7}, and \cot \theta=-\frac{7}{2} \.

Step 1: Determine the values of sine and cosine

From the given information, \(\tan \theta=-\frac{2}{7}\). We can think of tangent as \(\frac{opposite}{adjacent}\) in a right triangle. So, we can form a right triangle with -2 on the opposite side and 7 on the adjacent side. By the Pythagorean theorem, the hypotenuse is \(\sqrt{(-2)^2 + 7^2} = \sqrt{53}\). Since \(\sin \theta > 0\) and tangent is negative, the angle must be in the second quadrant (where sine is positive and tangent is negative). Thus, we get \(\sin \theta= \frac{-2}{\sqrt{53}}\) and \(\cos \theta= \frac{7}{\sqrt{53}}\).

Step 2: Calculate the other trigonometric functions

Using the definitions of the other trigonometric functions in terms of sine and cosine, we can find their exact values. The secant of theta, \(\sec \theta\), is the reciprocal of cosine, so \(\sec \theta=\frac{\sqrt{53}}{7}\). The cosecant of theta, \(\csc \theta\), is the reciprocal of sine, so \(\csc \theta=-\frac{\sqrt{53}}{2}\). Finally, the cotangent of theta, \(\cot \theta\), is the reciprocal of tangent, so \(\cot \theta=-\frac{7}{2}\).

Step 3: Gather all the results

Based on the calculations, the values of the trigonometric functions are:\( \sin \theta=\frac{-2}{\sqrt{53}} \), \( \cos \theta=\frac{7}{\sqrt{53}} \), \( \tan \theta=-\frac{2}{7} \),\( \csc \theta=-\frac{\sqrt{53}}{2} \), \( \sec \theta=\frac{\sqrt{53}}{7} \), and \( \cot \theta=-\frac{7}{2} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relations between triangle angles and sides. In particular, it focuses on right-angled triangles. You can visualize these triangles to understand trigonometric concepts easily. Trigonometric functions like sine, cosine, and tangent help us express these relationships.

• Sine (\( \sin \theta \)): Represents the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): Represents the ratio of the opposite side to the adjacent side.

These functions are crucial for calculations involving angle measures and distances in both two-dimensional and three-dimensional spaces. Understanding which quadrant your angle belongs to is essential, as it determines the sign of each function. For example, \( \sin \theta \) is positive in the first and second quadrants, making it essential to verify in which quadrant your angle lies during calculations. Smaller calculations, like finding the exact value of any function, depend on knowing these foundational definitions.

Pythagorean theorem

The Pythagorean theorem is a mathematical principle that adds depth to trigonometry. The theorem states that in a right triangle, the square of the hypotenuse (\( c \)) is equal to the sum of the squares of the other two sides (\( a \) and \( b \)). This is expressed as \( a^2 + b^2 = c^2 \).

• This theorem helps in converting the known lengths of sides into a triangle's hypothetically unknown side when the triangle is a right one.
• This becomes particularly useful in deriving trigonometric function values like sine and cosine from a tangent when only some relationships are known.

In the given exercise, where \( \tan \theta = -\frac{2}{7} \) and \( \sin \theta > 0 \), the Pythagorean theorem assists in calculating the hypotenuse needed to find the sine and cosine. It shows how a geometric concept can help bridge gaps in trigonometric calculations, particularly when applied to coordinate geometry's quadrants.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide an extended view of trigonometric relationships. They are particularly useful when direct values of sine, cosine, or tangent are unavailable but their reciprocal counterparts are known.

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

These functions are not merely mathematical tools but pragmatic solutions when dealing with complex angle measurements or when simplifying expressions. In the context of the original exercise, these reciprocal functions help in finding values like \( \csc \theta, \sec \theta, \) and \( \cot \theta \) using \( \sin \theta \) and \( \tan \theta \), efficiently demonstrating the interconnectedness of trigonometric calculations.