Problem 21 - Precalculus : Building Concepts and Connections

Question

Use the given value of a trigonometric function of \(\theta\) to find the values of the other five trigonometric functions. Assume \(\theta\) is an acute angle. $$\cot \theta=1.5$$

Step-by-Step Solution

Answer

Trigonometric values are \(\sin \theta = 0.5556\), \(\cos \theta = 0.8333\), \(\tan \theta = 0.6667\), \(\csc \theta = 1.8\), \(\sec \theta = 1.2\), \(\cot \theta = 1.5\).

Step 1: Find the tangent

Use the reciprocal relation \(\tan \theta = \frac{1}{\cot \theta}\). Substituting the given value we get: \(\tan \theta = \frac{1}{1.5} = 0.6667\)

Step 2: Get the sine and cosine

The tangent could also be expressed as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Choose any right triangle such that the ratio of the side opposite to \(\theta\) (which we take as \(\sin \theta\)) to the side adjacent to \(\theta\) (which we take as \(\cos \theta\)) is 0.6667. One of the easiest choices is to let \(\sin \theta = 0.6667\) and \(\cos \theta = 1\). Check this using the Pythagorean theorem: \((0.6667)^2 + 1^2 = 1.4449\), so in order to have this equal to 1 (since \(1^2 = \sin^2 \theta + \cos^2 \theta\)) we have to divide both \(\sin \theta\) and \(\cos \theta\) by \(\sqrt{1.4449}\). We obtain: \(\sin\theta = \frac{0.6667}{\sqrt{1.4449}} = 0.5556\) and \(\cos \theta = \frac{1}{\sqrt{1.4449}} = 0.8333\)

Step 3: Determine the other functions

Once we know the sine and cosine we can find the other functions. The secant and cosecant are the reciprocals of the cosine and sine respectively, thus: \(\sec \theta = \frac{1}{\cos \theta} = 1.2\) and \(\csc \theta = \frac{1}{\sin \theta} = 1.8\). The cotangent is the reciprocal of the tangent thus \(\cot \theta = 1.5\) as given. And finally \(\tan \theta = 0.6667\) as found in Step 1.

Key Concepts

Cotangent

The cotangent function, often denoted as \(\cot\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, so for any angle \(\theta\), \(\cot\theta = \frac{1}{\tan\theta}\). Generally, \(\cot\theta\) describes the ratio of the adjacent side to the opposite side in a right-angled triangle.\
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When solving problems like the exercise above, understanding this reciprocal relationship is crucial. If you are given \(\cot\theta\), you can immediately find \(\tan\theta\) by taking its reciprocal. In the given exercise, \(\cot\theta = 1.5\) leads to \(\tan\theta = \frac{1}{1.5}\) or approximately 0.6667. This relationship guides us towards further solutions involving other trigonometric functions. To grasp cotangent fully, always remember to visualize it as part of the trigonometric function family, each relating an angle to sides of a right triangle.

Sine and Cosine

Sine and cosine functions, represented by \(\sin\) and \(\cos\), are essential components of trigonometry. Sine corresponds to the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. Conversely, cosine relates the length of the adjacent side to the hypotenuse.\
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These functions are fundamental in connecting angles with lengths in right-angle triangles and in all sorts of applications extending to waves, oscillations, and circular motion. Following the given exercise, once the tangent (\(\tan\theta\)) is known, one can find a right triangle where \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), which is the ratio of the opposite side over the adjacent side. If you arbitrarily assign values to \(\sin\theta\) and \(\cos\theta\) that maintain this ratio, as shown in the solution, you can then adjust them to satisfy the Pythagorean theorem, another key concept in our discussion.\
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Remembering their geometrical meaning helps to visualize and solve various trigonometric problems, reinforcing the interconnectedness of these functions.

Pythagorean Theorem

The Pythagorean theorem is an essential principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)): \(c^2 = a^2 + b^2\). For trigonometric functions, this principle translates to the identity \(\sin^2\theta + \cos^2\theta = 1\), where the sine represents one side and the cosine the other, while the hypotenuse is of unit length.\
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The Pythagorean theorem underpins not just geometry but also trigonometry by linking the trigonometric functions to the lengths of sides of triangles. In our exercise, after finding the ratio of sine to cosine using the tangent value, we needed to ensure the sine and cosine values conform to the Pythagorean identity. By doing so, we find the correct values for \(\sin\theta\) and \(\cos\theta\), which are crucial for calculating the other trigonometric functions. Mastering the Pythagorean theorem is a powerful tool in solving a multitude of problems involving right triangles and trigonometric concepts.