Problem 23 - Precalculus with Limits

Question

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. $ -\dfrac{13\pi}{12}

Step-by-Step Solution

Answer

After doing all the calculations, we get that \(\sin(-\frac{13\pi}{12}) = - \frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(-\frac{13\pi}{12}) = - \frac{\sqrt{6} + \sqrt{2}}{4}\) and \(\tan(-\frac{13\pi}{12}) = 1 - \sqrt{3}\).

Step 1: Angle Conversion

The first stage is to convert the angle to one that is easily recognizable on the unit circle. The given angle can be written as a sum or difference of familiar angles. We can express \(-\frac{13\pi}{12}\) as \(-\frac{4\pi}{12} - \frac{9\pi}{12}\), which simplifies to \(-\frac{\pi}{3} - \frac{3\pi}{4}\).

Step 2: Finding Sine, Cosine and Tangent

The formulae for sine, cosine and tangent of the sum or difference of two angles must be used at this stage.\nFor sine: \(\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)\).\nFor cosine: \(\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)\).\nFor tangent: \(\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}\).\nNow we can plug in the angle values \(-\frac{\pi}{3}\) and \(-\frac{3\pi}{4}\) into these formulae, using standard trigonometric values for the familiar angles.

Step 3: Calculating Exact Values

Plug the familiar angle values into the formulae. From the unit circle, we know that \(\sin(-\frac{\pi}{3}) = -\sqrt{3}/2, \cos(-\frac{\pi}{3}) = 1/2, \tan(-\frac{\pi}{3}) = \sqrt{3}\) and \(\sin(-\frac{3\pi}{4}) = \sqrt{2}/2, \cos(-\frac{3\pi}{4}) = -\sqrt{2}/2, \tan(-\frac{3\pi}{4}) = -1\). Following the previous step, we substitute these values into our formulae to get the exact values for sine, cosine and tangent of the given angle.

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry. It helps us easily find the values of sine, cosine, and tangent for various angles. Imagine a circle with a radius of 1, located in a coordinate plane where the center of the circle is at the origin (0,0). Every angle on this circle corresponds to a point on the circle, giving us useful coordinates.

The x-coordinate of any point on this circle is the cosine of the angle, while the y-coordinate is the sine of the angle. This simple representation allows us to understand how these trigonometric functions relate to real numbers and angles.

• The circle’s circumference represents an angle in radians, making it easier to work with. For example, the rightmost point on this circle, corresponding to angle 0 radians, is (1,0).
• If you move counter-clockwise around the circle, each quarter (or 90°) splits the circle into four parts: first quadrant (0 to π/2), second quadrant (π/2 to π), third quadrant (π to 3π/2), and fourth quadrant (3π/2 to 2π).

The unit circle isn’t just a magic circle filled with numbers. It's a visualization that connects angles with coordinates, allowing us to look at and quickly remember the trigonometric values of common angles like π/3, π/4, and π/6.

Sum and Difference Identities

Sum and difference identities are formulas that allow us to find the sine, cosine, and tangent of the sum or difference of two angles. These identities are vital because they extend our ability to compute angles beyond those directly on the unit circle using known angles.

Specifically, with sum and difference identities, we can break complex angles down into parts—simpler angles that we know the values for. Here are the identities one might use:

• For sine: \(\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)\)
• For cosine:\(\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)\)
• For tangent:\(\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}\)

These formulas give us the exact values for angles like \(-\frac{13\pi}{12}\) by expressing them as the difference of two known angles, such as \(-\frac{\pi}{3}\) and \(-\frac{3\pi}{4}\), as demonstrated in the given problem.The understanding and application of these identities allow us to find precise trigonometric values without a calculator, using only the known values at familiar angles.

Exact Values of Sine, Cosine, Tangent

Knowing the exact values of sine, cosine, and tangent for standard angles is crucial for solving trigonometric problems effectively. The values are consistent and can be determined using the unit circle or through memorization of common angle values.

Consider angles like \(0, \pi/6, \pi/4, \pi/3, \text{ and } \pi/2\), whose trigonometric values we often learn through patterns and relationships within the unit circle.

• For example, at \(\pi/4\) (or 45°), both sine and cosine equal \(\frac{\sqrt{2}}{2}\).
• Many of these values form symmetrical patterns, like how the sine of \(\pi/3\) (or 60°) equals the cosine of \(\pi/6\) (or 30°).

By memorizing these patterns, we can easily plug known values into sum and difference identities to calculate the exact values for more complex angles.

In our exercise, by splitting \(-\frac{13\pi}{12}\) into parts and using these precise values — \(\sin\left(-\frac{\pi}{3}\right), \cos\left(-\frac{3\pi}{4}\right)\), etc. — one can determine the exact values of sine, cosine, and tangent at the original angle.