Problem 50
Question
Use a sketch to find the exact value of each expression. $$ \cot \left(\sin ^{-1} \frac{5}{13}\right) $$
Step-by-Step Solution
Verified Answer
\(\cot \left(\sin^{-1} \frac{5}{13}\right) = \frac{12}{5}\)
1Step 1: Draw a right triangle
Draw a right triangle such that the opposite side is 5 and the hypotenuse is 13. This corresponds to the value of \(\sin^{-1} \frac{5}{13}\). As sine is positive in this case and \(\sin^{-1} x\) is defined to yield values in the first and fourth quadrants (where \(y\) is positive or 0), we're dealing with a right triangle in the first quadrant, with angle \(\theta\) as the angle opposing the side of length 5.
2Step 2: Use Pythagorean theorem to find the adjacent side of the triangle
The Pythagorean theorem says that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, \(13^2 = 5^2 + a^2\) where \(a\) represents the adjacent side. Solving for \(a\) yields \(a = \sqrt{13^2 - 5^2} = 12\).
3Step 3: Use the definition of the cotangent function and substitute the triangle values
The cotangent of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the opposite side. Thus, \(\cot(\sin^{-1} \frac{5}{13}) = \cot(\theta) = \frac{Adjacent Side}{Opposite Side} = \frac{12}{5}\).
Key Concepts
Right TrianglePythagorean TheoremInverse Trigonometric FunctionsTrigonometric Ratios
Right Triangle
A right triangle is a triangle that features a 90-degree angle, also known as a right angle. This special type of triangle is foundational in trigonometry and allows for defining relationships between its three sides. In a right triangle, the side opposite the right angle is known as the hypotenuse, while the other two sides are often referred to as the adjacent and opposite sides relative to an angle of interest, usually denoted as \( \theta \).
The characteristics of right triangles enable the use of various trigonometric functions to solve problems related to angles and side lengths. When dealing with these triangles, the angles are generally measured in degrees or radians. The presence of a right angle simplifies many calculations thanks to established relationships such as trigonometric ratios and the Pythagorean theorem.
The characteristics of right triangles enable the use of various trigonometric functions to solve problems related to angles and side lengths. When dealing with these triangles, the angles are generally measured in degrees or radians. The presence of a right angle simplifies many calculations thanks to established relationships such as trigonometric ratios and the Pythagorean theorem.
Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in mathematics, particularly in the study of right triangles. It states that for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, this can be expressed as:
\[c^2 = a^2 + b^2\]
Here, \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides.
Using the Pythagorean theorem for a triangle where the hypotenuse is 13 and one of the sides (opposite) is 5, we can solve for the remaining side (adjacent). By substituting these values into the equation: \[13^2 = 5^2 + a^2\]Solving this equation, we find that the adjacent side length is 12, confirming the side lengths required for defining our trigonometric ratios.
\[c^2 = a^2 + b^2\]
Here, \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides.
Using the Pythagorean theorem for a triangle where the hypotenuse is 13 and one of the sides (opposite) is 5, we can solve for the remaining side (adjacent). By substituting these values into the equation: \[13^2 = 5^2 + a^2\]Solving this equation, we find that the adjacent side length is 12, confirming the side lengths required for defining our trigonometric ratios.
Inverse Trigonometric Functions
Inverse trigonometric functions are utilized to determine the values of angles within triangles when given a ratio of sides. These functions are the inverses of the basic trigonometric functions (sine, cosine, and tangent) and are crucial in solving triangles.
The notation for these functions typically includes an arc prefix (like arcsin, arccos, arctan) or a superscript \(^{-1}\), indicating the inverse function.
For the exercise at hand, \(\sin^{-1}\left(\frac{5}{13}\right)\) helps us find the angle \(\theta\) with a sine ratio of \(\frac{5}{13}\).
With inverse functions, it's important to recognize their ranges. \(\sin^{-1}(x)\) results in angles from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), meaning the angle lies in the first or fourth quadrants. This helps determine orientation and side relationships in the right triangle.
The notation for these functions typically includes an arc prefix (like arcsin, arccos, arctan) or a superscript \(^{-1}\), indicating the inverse function.
For the exercise at hand, \(\sin^{-1}\left(\frac{5}{13}\right)\) helps us find the angle \(\theta\) with a sine ratio of \(\frac{5}{13}\).
With inverse functions, it's important to recognize their ranges. \(\sin^{-1}(x)\) results in angles from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), meaning the angle lies in the first or fourth quadrants. This helps determine orientation and side relationships in the right triangle.
Trigonometric Ratios
Trigonometric ratios play an essential role in understanding the relationships between the angles and sides of right triangles. These ratios include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), along with their reciprocals cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).
Each ratio is derived from the sides of a right triangle relative to an angle \(\theta\):
Each ratio is derived from the sides of a right triangle relative to an angle \(\theta\):
- Sine (\(\sin\)) is the ratio of the opposite side to the hypotenuse.
- Cosine (\(\cos\)) is the ratio of the adjacent side to the hypotenuse.
- Tangent (\(\tan\)) is the ratio of the opposite side to the adjacent side.
- Cotangent (\(\cot\)) is the reciprocal of tangent, or the ratio of the adjacent side to the opposite side.
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