Problem 69
Question
Find the indicated trigonometric value in the specified quadrant. $$ \sin \theta=-\frac{3}{5} \quad \text { IV } \quad \cos \theta $$
Step-by-Step Solution
Verified Answer
The value of \(\cos \theta\) in the fourth quadrant, when \(\sin \theta = -3/5\), is +4/5.
1Step 1: Identify the given value and quadrant
The given value for \(\sin \theta\) is -3/5 and the provided quadrant is IV.
2Step 2: Use the Pythagorean trigonometric identity formula
Reshape the Pythagorean trigonometric identity formula and isolate \(\cos \theta\). Formula becomes: \( \cos^2\theta = 1 - \sin^2\theta \).
3Step 3: Insert the given value in the formula
Insert the value of \(\sin \theta\) into the formula from the last step and calculate the value of variable under the square root. Calculate \( 1 - (-3/5)^2\) to get \( 1 - 9/25 = 16/25 \). The root of this is ±4/5.
4Step 4: Identify the correct sign of the answer
Since in the fourth quadrant, cosine is positive, therefore, the solution \(\cos\theta\) is +4/5.
Key Concepts
Pythagorean Trigonometric IdentitySine and CosineQuadrants of the Coordinate System
Pythagorean Trigonometric Identity
One of the most fundamental concepts in trigonometry is the Pythagorean trigonometric identity. It is an equation that relates the square of the sine and cosine of an angle. This identity is derived from the Pythagorean Theorem, which you may know from geometry, involving the sides of a right triangle.
The identity states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) equals 1: \[\sin^2\theta + \cos^2\theta = 1\].
In the exercise, we use this identity to find \( \cos \theta \) given \( \sin \theta \). We begin by subtracting \( \sin^2\theta \) from both sides of the identity. This leaves us with \(\cos^2\theta = 1 - \sin^2\theta\), which we can then solve for \( \cos \theta \) by taking the square root. Since the square root has both a positive and negative solution, we use additional information about the angle's quadrant to determine the correct sign for \( \cos \theta \).
The identity states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) equals 1: \[\sin^2\theta + \cos^2\theta = 1\].
In the exercise, we use this identity to find \( \cos \theta \) given \( \sin \theta \). We begin by subtracting \( \sin^2\theta \) from both sides of the identity. This leaves us with \(\cos^2\theta = 1 - \sin^2\theta\), which we can then solve for \( \cos \theta \) by taking the square root. Since the square root has both a positive and negative solution, we use additional information about the angle's quadrant to determine the correct sign for \( \cos \theta \).
Sine and Cosine
Sine and cosine are the basic trigonometric functions that associate an angle of a right triangle with the ratio of two sides.
The sine of an angle \( \theta \), denoted as \( \sin(\theta) \), is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, the cosine, denoted as \( \cos(\theta) \), is the ratio of the length of the adjacent side to the hypotenuse. They can be remembered easily as \(\sin(\theta) = \frac{opposite}{hypotenuse}\) and \(\cos(\theta) = \frac{adjacent}{hypotenuse}\).
Because these functions are based on ratios, they can be defined for any size of a right triangle and thus for any real number as an angle by considering a unit circle. The values of sine and cosine range from -1 to 1, inclusive. In the provided exercise, we're given the sine value and need to deduce the cosine value, which requires understanding the relationship between these two functions as expressed by the Pythagorean identity.
The sine of an angle \( \theta \), denoted as \( \sin(\theta) \), is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, the cosine, denoted as \( \cos(\theta) \), is the ratio of the length of the adjacent side to the hypotenuse. They can be remembered easily as \(\sin(\theta) = \frac{opposite}{hypotenuse}\) and \(\cos(\theta) = \frac{adjacent}{hypotenuse}\).
Because these functions are based on ratios, they can be defined for any size of a right triangle and thus for any real number as an angle by considering a unit circle. The values of sine and cosine range from -1 to 1, inclusive. In the provided exercise, we're given the sine value and need to deduce the cosine value, which requires understanding the relationship between these two functions as expressed by the Pythagorean identity.
Quadrants of the Coordinate System
The coordinate system is divided into four parts called quadrants, which are counterclockwise from the positive x-axis. They're numbered as Quadrant I, II, III, and IV. In each of these quadrants, the signs of sine and cosine functions vary.
In Quadrant I, both sine and cosine values are positive. Moving to Quadrant II, sine stays positive but cosine becomes negative. In Quadrant III, sine values are negative and cosine are also negative. Lastly, in Quadrant IV, where our exercise angle is located, sine is negative and cosine is positive.
This knowledge is essential when solving trigonometric problems because after using identities to find the value of a function, the quadrant tells us whether that value should be considered positive or negative. As in our exercise, knowing that \( \sin \theta \) is negative in Quadrant IV and since the cosine is positive in this quadrant, we correctly concluded that \( \cos \theta \) had to be the positive value of the square root, giving us \(+\frac{4}{5}\) as the solution.
In Quadrant I, both sine and cosine values are positive. Moving to Quadrant II, sine stays positive but cosine becomes negative. In Quadrant III, sine values are negative and cosine are also negative. Lastly, in Quadrant IV, where our exercise angle is located, sine is negative and cosine is positive.
This knowledge is essential when solving trigonometric problems because after using identities to find the value of a function, the quadrant tells us whether that value should be considered positive or negative. As in our exercise, knowing that \( \sin \theta \) is negative in Quadrant IV and since the cosine is positive in this quadrant, we correctly concluded that \( \cos \theta \) had to be the positive value of the square root, giving us \(+\frac{4}{5}\) as the solution.
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