Problem 33
Question
The terminal side of \(\boldsymbol{\theta}\) lies on the given line in the specified quadrant. Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) by finding a point on the line. y=-x, II
Step-by-Step Solution
Verified Answer
The six trigonometric functions for the point (-1,1) are: \(\sin(\theta) = \sqrt{2}/2\), \(\cos(\theta) = -\sqrt{2}/2\), \(\tan(\theta) = -1\), \(\csc(\theta) = \sqrt{2}\), \(\sec(\theta) = -\sqrt{2}\), \(\cot(\theta) = -1\)
1Step 1: Identifying a Suitable Point in the Second Quadrant on the Line y = -x
We can conveniently use the point (-1,1). It is in the second quadrant since the x-coordinate is negative and the y-coordinate is positive. In addition, this point lies on the line defined by the equation \(y=-x\). This point will also simplify our calculations since both the x- and y- coordinates are 1 in magnitude.
2Step 2: Identify the Hypotenuse of the Right Triangle
In the right triangle with a horizontal coordinate of -1 and a vertical coordinate of 1, the hypotenuse, per Pythagoras theorem, is given as \(r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\).
3Step 3: Calculate the Trigonometric Functions
The six trigonometric functions for the defined point at (\(-1\), \(1\)) and hypotenuse of \(\sqrt{2}\) can be computed as follows:\n\nSine \(\sin(\theta) = 1/r = 1/\sqrt{2} = \sqrt{2}/2,\)\n\nCosine: \(\cos(\theta) = -1/r = -1/\sqrt{2} = -\sqrt{2}/2,\)\n\nTangent: \(\tan(\theta) = y/x = 1/-1 = -1,\)\n\nCosecant: \(\csc(\theta) = r/y = \sqrt{2},\)\n\nSecant: \(\sec(\theta) = r/x = -\sqrt{2},\)\n\nCotangent: \(\cot(\theta) = x/y = -1.\)
Key Concepts
Right Triangle TrigonometryUnit CirclePythagorean Theorem
Right Triangle Trigonometry
Trigonometry is the study of the relationships involving the lengths and angles of triangles, especially right triangles. In the context of right triangle trigonometry, we use specific trigonometric functions to relate the angles to the side lengths. Each function provides a different piece of information about the triangle. Imagine a right triangle where one angle, which we call \(\theta\), is known, and the lengths of the sides opposite and adjacent to \(\theta\) are also given.
For such a triangle, the sine function \(\sin(\theta)\) tells us the ratio of the length of the side opposite \(\theta\) to the hypotenuse. The cosine function \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse, while the tangent function \(\tan(\theta)\) is the ratio of the opposite side to the adjacent side. These are the primary or basic trigonometric functions, but there are also the reciprocal functions: cosecant \(\csc(\theta)\), secant \(\sec(\theta)\), and cotangent \(\cot(\theta)\), which are the reciprocals of sine, cosine, and tangent, respectively.
Using the right triangle trigonometry, we can solve a wealth of problems by just knowing an angle and one side length. The usefulness of this principle is evident in real-life applications such as in engineering, physics, and even in determining heights and distances that would be otherwise difficult to measure.
For such a triangle, the sine function \(\sin(\theta)\) tells us the ratio of the length of the side opposite \(\theta\) to the hypotenuse. The cosine function \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse, while the tangent function \(\tan(\theta)\) is the ratio of the opposite side to the adjacent side. These are the primary or basic trigonometric functions, but there are also the reciprocal functions: cosecant \(\csc(\theta)\), secant \(\sec(\theta)\), and cotangent \(\cot(\theta)\), which are the reciprocals of sine, cosine, and tangent, respectively.
Using the right triangle trigonometry, we can solve a wealth of problems by just knowing an angle and one side length. The usefulness of this principle is evident in real-life applications such as in engineering, physics, and even in determining heights and distances that would be otherwise difficult to measure.
Unit Circle
The unit circle is a fundamental concept in trigonometry and appears when we extend trigonometric functions to all real numbers, not just those corresponding to angles of a right triangle. A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle \(\theta\) that the point makes with the positive x-axis, and the y-coordinate corresponds to the sine of that angle.
Therefore, any point on the unit circle has coordinates \( (\cos(\theta), \sin(\theta)) \), which means that the trigonometric functions can now be extended to angles greater than \(90^\circ\) (which is not possible in the context of right triangles). This strategy allows us to find the trigonometric values for angles located in any of the four quadrants of the coordinate plane. For instance, in the solution mentioned above, the angle \(\theta\) leads us to a point in the second quadrant where the sine is positive and the cosine is negative. It is an elegant way of visualizing and working with angles and their associated trigonometric functions.
Therefore, any point on the unit circle has coordinates \( (\cos(\theta), \sin(\theta)) \), which means that the trigonometric functions can now be extended to angles greater than \(90^\circ\) (which is not possible in the context of right triangles). This strategy allows us to find the trigonometric values for angles located in any of the four quadrants of the coordinate plane. For instance, in the solution mentioned above, the angle \(\theta\) leads us to a point in the second quadrant where the sine is positive and the cosine is negative. It is an elegant way of visualizing and working with angles and their associated trigonometric functions.
Pythagorean Theorem
The Pythagorean Theorem is a well-established principle in mathematics, particularly in geometry, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is written as \( a^2 + b^2 = c^2 \), where \(c\) is the hypotenuse, while \(a\) and \(b\) are the lengths of the triangle's other two sides. The theorem is not only a cornerstone in the study of geometry but is also instrumental in various areas of science, engineering, and different branches of mathematics.
Referring back to the provided exercise, the solution employs the Pythagorean Theorem to identify the length of the hypotenuse of the right triangle formed by the point (-1,1) on the unit circle. By squaring the x-coordinate and y-coordinate and taking the square root of their sum, we determine the hypotenuse. The theorem's simple formula provides a powerful tool for solving many practical problems, especially when combined with trigonometry to find distances or to navigate between coordinate points.
Referring back to the provided exercise, the solution employs the Pythagorean Theorem to identify the length of the hypotenuse of the right triangle formed by the point (-1,1) on the unit circle. By squaring the x-coordinate and y-coordinate and taking the square root of their sum, we determine the hypotenuse. The theorem's simple formula provides a powerful tool for solving many practical problems, especially when combined with trigonometry to find distances or to navigate between coordinate points.
Other exercises in this chapter
Problem 33
Sketch the graph of the function. Include two full periods. $$ y=2 \csc (x-\pi) $$
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Graph \(f\) and \(g\) on the same set of coordinate axes. (Include two full periods.) $$ \begin{array}{l} f(x)=\cos x \\ g(x)=2+\cos x \end{array} $$
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Use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. \(\cos \thet
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Evaluate (if possible) the six trigonometric functions of the real number. $$ t=-\frac{\pi}{2} $$
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