Problem 28 - Contemporary Precalculus

Question

Simplify the given expression. $$\text { If } \sin x=-\frac{3}{4} \text { and } \frac{3 \pi}{2}

Step-by-Step Solution

Answer

Answer: \(\frac{\sqrt{14} + 3\sqrt{2}}{8}\)

Step 1: Determine Quadrant

Given that \(\frac{3 \pi}{2}

Step 2: Calculate \(\cos x\)

Since x is in the 4th quadrant and we know that \(\sin x = -\frac{3}{4}\), we can use the Pythagorean identity for sine and cosine: $$\sin^2{x}+\cos^2{x} = 1$$ Substitute the given value of \(\sin x\): $$\left(-\frac{3}{4}\right)^2+\cos^2{x} = 1$$ Solve for \(\cos x\): $$\frac{9}{16}+\cos^2{x} = 1$$ $$\cos^2{x} = 1-\frac{9}{16} = \frac{7}{16}$$ $$\cos x = \pm\frac{\sqrt{7}}{4}$$ In the 4th quadrant, cosine is positive, so: $$\cos x = \frac{\sqrt{7}}{4}$$

Step 3: Use the angle addition formula

Now we use the angle addition formula for cosine to find \(\cos\left(\frac{\pi}{4}+x\right)\): $$\cos\left(\frac{\pi}{4}+x\right) = \cos\frac{\pi}{4}\cos x - \sin \frac{\pi}{4} \sin x$$

Step 4: Substitute values and solve

Substitute the values of \(\sin x\), \(\cos x\), \(\sin \frac{\pi}{4}\), and \(\cos \frac{\pi}{4}\), and solve for the expression: $$\cos \left(\frac{\pi}{4}+x\right) = \cos\left(\frac{\pi}{4}\right) \cdot \frac{\sqrt{7}}{4} - \sin\left(\frac{\pi}{4}\right) \cdot -\frac{3}{4}$$ Since \(\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\), we get: $$\cos \left(\frac{\pi}{4}+x\right) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{7}}{4} - \frac{\sqrt{2}}{2} \cdot -\frac{3}{4}$$ Finally, simplify the expression: $$\cos \left(\frac{\pi}{4}+x\right) = \frac{\sqrt{14}}{8} + \frac{3\sqrt{2}}{8} = \frac{\sqrt{14} + 3\sqrt{2}}{8}$$ So, the simplified expression for \(\cos \left(\frac{\pi}{4}+x\right)\) is \(\frac{\sqrt{14} + 3\sqrt{2}}{8}\).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying trigonometric expressions and solving various math problems. These identities are equations that involve trigonometric functions and are true for all values of the involved variables. A fundamental example is the Pythagorean identity: \[ \text{sin}^2{theta} + \text{cos}^2{theta} = 1 \] This particular identity is derived from the Pythagorean theorem and relates the sine and cosine functions of an angle. It's utilized when we have one trigonometric function (sin or cos) and need to find another.

In the exercise, knowing the Pythagorean identity allowed us to solve for \text{cos} x given the \text{sin} x value, by rearranging the identity. Since we were given that \text{sin} x = -\frac{3}{4}, we simply squared this value and subtracted it from 1 to find \text{cos}^2{ x}, and then took the square root, considering the quadrant to determine the sign. Accurate use of these identities is critical for students to reach the right solutions.

Angle Addition Formulas

Angle addition formulas allow us to find the sine, cosine, and tangent of the sum or difference of two angles. For instance, the angle addition formula for cosine is: \[ \text{cos}(a+b) = \text{cos} a \times \text{cos} b - \text{sin} a \times \text{sin} b \] These are extremely useful when dealing with expressions involving the sum or difference of two angles. In our exercise, the angle addition formula was used to simplify \text{cos} \left(\frac{\pi}{4}+x\right).

By strategically substituting the known values, \text{sin} x and \text{cos} x, as well as using the known values of \text{sin} \frac{\pi}{4} and \text{cos} \frac{\pi}{4}, students can find the exact trigonometric value of the combined angle. Mastering these formulas is crucial for progressing in trigonometry.

Quadrant Analysis

Quadrant analysis is a part of trigonometry where we determine the sign and sometimes the value of trigonometric functions based on the angle's quadrant. Angles are divided into four quadrants in the coordinate plane, with each quadrant demonstrating specific sign patterns for sine, cosine, and tangent functions.

For instance, in the first quadrant, all trigonometric values are positive. Moving to the fourth quadrant, where our given angle x lies (since \(\frac{3 \pi}{2}Working Out SignsThe correct sign of the trigonometric functions is crucial for finding the actual values, as seen in our exercise. Given \(\sin x=-\frac{3}{4}\), we knew that x is in the fourth quadrant; combined with the Pythagorean identity, we deduced that \(\cos x\) must be positive. Knowing which quadrant an angle is in, and the corresponding signs of the trigonometric functions, helps greatly in simplifying expressions and solving equations.

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Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying trigonometric expressions and solving various math problems. These identities are equations that involve trigonometric functions and are true for all values of the involved variables. A fundamental example is the Pythagorean identity: \[ \text{sin}^2{theta} + \text{cos}^2{theta} = 1 \] This particular identity is derived from the Pythagorean theorem and relates the sine and cosine functions of an angle. It's utilized when we have one trigonometric function (sin or cos) and need to find another.

In the exercise, knowing the Pythagorean identity allowed us to solve for \text{cos} x given the \text{sin} x value, by rearranging the identity. Since we were given that \text{sin} x = -\frac{3}{4}, we simply squared this value and subtracted it from 1 to find \text{cos}^2{ x}, and then took the square root, considering the quadrant to determine the sign. Accurate use of these identities is critical for students to reach the right solutions.

Angle Addition Formulas

Angle addition formulas allow us to find the sine, cosine, and tangent of the sum or difference of two angles. For instance, the angle addition formula for cosine is: \[ \text{cos}(a+b) = \text{cos} a \times \text{cos} b - \text{sin} a \times \text{sin} b \] These are extremely useful when dealing with expressions involving the sum or difference of two angles. In our exercise, the angle addition formula was used to simplify \text{cos} \left(\frac{\pi}{4}+x\right).

By strategically substituting the known values, \text{sin} x and \text{cos} x, as well as using the known values of \text{sin} \frac{\pi}{4} and \text{cos} \frac{\pi}{4}, students can find the exact trigonometric value of the combined angle. Mastering these formulas is crucial for progressing in trigonometry.

Quadrant Analysis

Quadrant analysis is a part of trigonometry where we determine the sign and sometimes the value of trigonometric functions based on the angle's quadrant. Angles are divided into four quadrants in the coordinate plane, with each quadrant demonstrating specific sign patterns for sine, cosine, and tangent functions.

For instance, in the first quadrant, all trigonometric values are positive. Moving to the fourth quadrant, where our given angle x lies (since \(\frac{3 \pi}{2}Working Out SignsThe correct sign of the trigonometric functions is crucial for finding the actual values, as seen in our exercise. Given \(\sin x=-\frac{3}{4}\), we knew that x is in the fourth quadrant; combined with the Pythagorean identity, we deduced that \(\cos x\) must be positive. Knowing which quadrant an angle is in, and the corresponding signs of the trigonometric functions, helps greatly in simplifying expressions and solving equations.