Problem 70
Question
Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sec \left[\arctan \left(-\frac{3}{5}\right)\right]\)
Step-by-Step Solution
Verified Answer
The exact value of the expression \(\sec[\arctan(-\frac{3}{5})]\) is \(\frac{\sqrt{34}}{5}\).
1Step 1: Draw a Right Triangle and Assign Values
Draw a right triangle where the opposite side (height) is -3 (denoting downward direction) and the adjacent side (base) is 5. The angle we're interested in is the one whose tangent is -3/5. That angle is going to be \(\arctan(-\frac{3}{5})\). The hypotenuse of this right triangle using Pythagorean theorem is \(\sqrt{(-3)^2+(5)^2}=\sqrt{34}\).
2Step 2: Calculate the Secant
Secant (\(\sec\)) is the reciprocal of the cosine function. Looking at the triangle, cosine of the angle is given by adjacent side divided by the hypotenuse. Therefore, \(\cos[\arctan(-\frac{3}{5})]=\frac{5}{\sqrt{34}}\). From this, we find that the secant is: \(\sec[\arctan(-\frac{3}{5})]=\frac{1}{\cos[\arctan(-\frac{3}{5})]}=\frac{\sqrt{34}}{5}\).
3Step 3: Use a Graphing Utility for Verification
Once you obtain the exact value for the secant, use a graphing utility to verify your result. Input the original expression \(\sec[\arctan(-\frac{3}{5})]\) into the graphing utility and it should return a decimal equivalent of your exact value \(\frac{\sqrt{34}}{5}\).
Key Concepts
Right Triangle TrigonometryPythagorean TheoremGraphing Utilities VerificationInverse Trigonometric Functions
Right Triangle Trigonometry
Right triangle trigonometry is a component of geometry where we study the relationships between angles and sides of right triangles. A right triangle, as its name suggests, includes one angle that is precisely 90 degrees, known as the right angle. The sides of the triangle have special names: the side opposite the right angle is the 'hypotenuse', the side opposite the angle of interest is known as the 'opposite' side, and the remaining side is called the 'adjacent' side.
In the context of the exercise, we're dealing with the tangent of an angle, which is defined as the ratio of the length of the opposite side to the length of the adjacent side. When the tangent is negative, as in \(\tan[\text{angle}] = -\frac{3}{5}\), this implies that the angle is located in either the second or fourth quadrant, where the tangent values are negative.
Understanding these relationships and labels is key in solving trigonometric problems involving right triangles and is especially useful when we want to determine the exact value of trigonometric functions, such as in the given exercise with the secant (reciprocal of cosine) of the arctangent of a ratio.
In the context of the exercise, we're dealing with the tangent of an angle, which is defined as the ratio of the length of the opposite side to the length of the adjacent side. When the tangent is negative, as in \(\tan[\text{angle}] = -\frac{3}{5}\), this implies that the angle is located in either the second or fourth quadrant, where the tangent values are negative.
Understanding these relationships and labels is key in solving trigonometric problems involving right triangles and is especially useful when we want to determine the exact value of trigonometric functions, such as in the given exercise with the secant (reciprocal of cosine) of the arctangent of a ratio.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry, stating that in a right triangle, the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\): \(c^2 = a^2 + b^2\).
In our exercise, we apply this theorem to find the length of the hypotenuse, given the lengths of the opposite and adjacent sides. With the opposite side being -3, and the adjacent side 5, the length of the hypotenuse is the square root of the sum of the squares of both sides: \(\sqrt{(-3)^2 + (5)^2} = \sqrt{34}\).
This theorem is critical in trigonometry as it provides a means to calculate the lengths of sides when angles and other sides are known, which is particularly useful when solving problems that include trigonometric functions.
In our exercise, we apply this theorem to find the length of the hypotenuse, given the lengths of the opposite and adjacent sides. With the opposite side being -3, and the adjacent side 5, the length of the hypotenuse is the square root of the sum of the squares of both sides: \(\sqrt{(-3)^2 + (5)^2} = \sqrt{34}\).
This theorem is critical in trigonometry as it provides a means to calculate the lengths of sides when angles and other sides are known, which is particularly useful when solving problems that include trigonometric functions.
Graphing Utilities Verification
Graphing utilities verification involves using technology, like graphing calculators or software, to confirm the results of our trigonometric calculations. These tools are capable of plotting equations, executing complex functions, and providing visualization which aids our understanding of the concepts.
To verify our secant of arctangent calculation, we input the expression \(\sec[\arctan(-\frac{3}{5})]\) into the graphing utility. The tool can then provide a decimal approximation, which we can compare to our exact answer of \(\frac{\sqrt{34}}{5}\) to check correctness. This step is a practical application of technology in mathematics, which not only confirms our solution but can also help in visual learning and understanding of the adaptations of trigonometric functions on a graph.
To verify our secant of arctangent calculation, we input the expression \(\sec[\arctan(-\frac{3}{5})]\) into the graphing utility. The tool can then provide a decimal approximation, which we can compare to our exact answer of \(\frac{\sqrt{34}}{5}\) to check correctness. This step is a practical application of technology in mathematics, which not only confirms our solution but can also help in visual learning and understanding of the adaptations of trigonometric functions on a graph.
Inverse Trigonometric Functions
Inverse trigonometric functions, also called arc functions, help us find an angle when the value of a trigonometric function is known. The common inverse trigonometric functions include arcsine (\(\arcsin\)), arccosine (\(\arccos\)), and arctangent (\(\arctan\)).
In the exercise, the arctangent of -3/5, denoted by \(\arctan(-\frac{3}{5})\), is used to find the angle whose tangent is -3/5. The secant of this angle is then calculated. The concept of inverse functions is crucial in trigonometry because it allows us to work backwards from the ratio of sides back to the angle, completing our understanding of the interplay between angles and sides in trigonometric functions.
In the exercise, the arctangent of -3/5, denoted by \(\arctan(-\frac{3}{5})\), is used to find the angle whose tangent is -3/5. The secant of this angle is then calculated. The concept of inverse functions is crucial in trigonometry because it allows us to work backwards from the ratio of sides back to the angle, completing our understanding of the interplay between angles and sides in trigonometric functions.
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