Problem 67 - 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) \(\sin \theta=\frac{1}{2}\) (b) \(\csc \theta=2\)

Step-by-Step Solution

Answer

For both \(\sin \theta = \frac{1}{2}\) and \(\csc \theta = 2\), \(\theta = 30^{\circ}\) or \(\frac{\pi}{6}\) radians.

Step 1: Identify the corresponding angle for \(\sin \theta = \frac{1}{2}\)

From the trigonometric values that are assumed to be known or remembered, it is known that \(\sin \theta = \frac{1}{2}\) for \(\theta = 30^{\circ}\) or \(\frac{\pi}{6}\).

Step 2: Determine the value of \(\theta\) for \(\csc \theta = 2\)

Cosecant is reciprocal of sine. Hence, \(\csc \theta = 2\) means that, \(\sin \theta = \frac{1}{2}\). Therefore, using similar logic as in Step 1, it can be seen that \(\csc \theta = 2\) for \(\theta = 30^{\circ}\) or \(\frac{\pi}{6}\).

Key Concepts

Sine

Understanding sine in trigonometry is fundamental to grasping the relationships between angles and sides of a right triangle. In a right-angled triangle, the sine of an angle \theta, written as \(\sin \theta\), is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse (the longest side of the triangle).
For example, when dealing with common angles such as \(30^\circ\) or \(45^\circ\), we often memorize their sine values. The sine of \(30^\circ\), which translates to \(\frac{\pi}{6}\) in radians, is \(\frac{1}{2}\). This means if you have a right triangle with a \(30^\circ\) angle, the side opposite this angle is half the length of the hypotenuse.\
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Memorizing certain sine values for common angles can be incredibly helpful, especially when you're without a calculator. For instance, the sine values for \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) are easy to remember and can assist you in solving trigonometric problems quickly.

Cosecant

The cosecant function, abbreviated as \(\csc\), may not be as familiar as sine, but it is simply the reciprocal of the sine function. So, \(\csc \theta = \frac{1}{\sin \theta}\).
Working with this function does not have to be difficult when you remember that it is just 'flipping' the sine value. If you know that \(\sin \theta\) is \(\frac{1}{2}\), then it follows that \(\csc \theta\) will be \(2\). This comes in handy when you do not have access to a calculator; knowing one value allows you to find the other quickly.
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Understanding the reciprocal nature of sine and cosecant is essential. It's worth keeping in mind the major cosecant values that correspond to the common angles, as this knowledge can simplify complex problems to simple, solvable components.

Radians in Trigonometry

Radians offer a different way to measure angles as opposed to degrees. In the context of trigonometry, understanding radians is just as important as understanding degrees. One complete revolution around a circle is \(2\pi\) radians, which is equal to \(360^\circ\). This means that \(\pi\) radians is \(180^\circ\), making it relatively easy to convert between the two.
Using radians can sometimes simplify the mathematics involved in trigonometric calculations, especially when dealing with periodic functions like sine and cosine. Radians are also more natural when it comes to calculus and mathematical analysis, making it a critical concept to understand in higher mathematics. Memorizing the radian equivalents of common trigonometric angles can help you swiftly navigate problems without reaching for a calculator.

Trigonometric Angles

Trigonometric angles are the angles for which we can easily recall the sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent) values. These are typically the angles that form part of a special right triangle - usually \(30^\circ\text{-}60^\circ\)- or \(45^\circ\) triangles - or are based on the unit circle where the radius is one.
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Knowing the exact values of the trigonometric functions for these angles lets us solve trigonometry problems more efficiently. A solid grasp of trigonometric angles and their corresponding values is a powerful tool in a mathematician's toolkit, especially in situations where technology is inaccessible. For example, with the trigonometric angles, if you know that the sine of \(30^\circ\) is \(\frac{1}{2}\), you can immediately deduce that the cosine of \(60^\circ\) is also \(\frac{1}{2}\) due to the complementary nature of these angles in a right triangle.