Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\cot \theta=\frac{\sqrt{3}}{3}\) (b) \(\sec \theta=\sqrt{2}\)

Step-by-Step Solution

Answer

For \(\cot \theta=\frac{\sqrt{3}}{3}\), the solutions are \(60^\circ\) or \(\frac{\pi}{3}\) radians. For \(\sec \theta=\sqrt{2}\), the solutions are \(45^\circ\) or \(\frac{\pi}{4}\) radians.

Step 1: Solve for \(\theta\) when \(\cot \theta=\frac{\sqrt{3}}{3}\)

Recall that \(\cot \theta = \frac{1}{\tan \theta}\) . Thus, we have that \(\frac{1}{\tan \theta} = \frac{\sqrt{3}}{3}\), which implies \(\tan\theta = \frac{3}{\sqrt{3}} = \sqrt{3}\). The angle \(\theta\) in the first quadrant that gives \(\tan \theta = \sqrt{3}\) is \(60^\circ\) or \(\frac{\pi}{3}\) radians.

Step 2: Solve for \(\theta\) when \(\sec \theta=\sqrt{2}\)

Recall the definition of secant \(\sec \theta = \frac{1}{\cos \theta}\) . Thus, we have that \(\frac{1}{\cos \theta} = \sqrt{2}\), which implies \(\cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\). The angle \(\theta\) in the first quadrant that gives \(\cos \theta = \frac{\sqrt{2}}{2}\) is \(45^\circ\) or \(\frac{\pi}{4}\) radians.

Key Concepts

Cotangent

The cotangent is a trigonometric function that is the reciprocal of the tangent. If you understand tangent as \\( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), then cotangent \( \cot \theta \) is defined as \\( \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}} \). This makes cotangent very useful when the lengths of the adjacent and opposite sides of a right triangle are known.
When solving for \( \theta \) given \( \cot \theta = \frac{\sqrt{3}}{3} \), it's helpful to rewrite it in terms of tangent: \( \tan \theta = \sqrt{3} \). In the context of a unit circle and using angles in the first quadrant (0 to 90 degrees), the cotangent function value of \( \frac{\sqrt{3}}{3} \) corresponds to \( \theta = 60^\circ \) or \( \frac{\pi}{3} \) radians. Knowing this can solve trigonometry problems efficiently without a calculator by remembering a few angle-value pairs.

Secant

The secant function is another trigonometric ratio, specifically the reciprocal of cosine. \\( \sec \theta = \frac{1}{\cos \theta} \). This is particularly useful for situations where the cosine of an angle might be known or measured directly.
In practical terms, to find \( \theta \) when \( \sec \theta = \sqrt{2} \), we want to rewrite this in terms of cosine: \( \cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \). This aligns with one of the standard angles in trigonometry. Specifically, for an angle \( \theta \) in the first quadrant, \( \cos \theta = \frac{\sqrt{2}}{2} \) corresponds to \( \theta = 45^\circ \) or \( \frac{\pi}{4} \) radians. Remembering these relationships can simplify trigonometry problems significantly.

Angle Measurement

Angles can be measured in degrees or radians, two systems with their advantages. Degrees are often more intuitive, especially when learning basic trigonometric principles, because they break down full circles into 360 equal parts.
Simple angles like 30° or 90° are easier to visualize and recall. Radians, however, play a critical role in higher mathematics because they relate directly to the radius of a circle. A full circle is \( 2\pi \) radians, making it a naturally occurring unit in trigonometric functions.\

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• A right angle is 90 degrees or \( \frac{\pi}{2} \) radians.
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• A straight angle is 180 degrees or \( \pi \) radians.
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\Understanding these units facilitates transition between the two systems and is crucial in fields like calculus or when working with circular functions.

Degree-Radian Conversion

Converting between degrees and radians is an essential skill. It's grounded in the relationship of a complete circle's circumference in both units: 360 degrees equals \( 2\pi \) radians. To convert from degrees to radians, use the formula \\( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). For instance, to convert 60 degrees into radians, compute \\( 60 \times \frac{\pi}{180} = \frac{\pi}{3} \). To go from radians to degrees, the formula is reversed: \\( \text{degrees} = \text{radians} \times \frac{180}{\pi} \). An example is converting \( \frac{\pi}{4} \) radians to degrees: \\( \frac{\pi}{4} \times \frac{180}{\pi} = 45 \) degrees. Mastery of these conversions simplifies mathematical problem-solving, especially in contexts requiring both measurement formats.