Problem 2 - Calculus Single Variable

Question

Calculate each of the six trigonometric functions at angle \(\theta\) without using a calculator. \(\theta=\pi / 4\)

Step-by-Step Solution

Answer

For \( \theta = \frac{\pi}{4} \), the trigonometric functions are: \( \sin = \cos = \frac{\sqrt{2}}{2}\), \( \tan = 1\), \( \csc = \sec = \sqrt{2} \), \( \cot = 1 \).

Step 1: Understand the Angle

The angle \( \theta = \frac{\pi}{4} \) is a common angle used in trigonometry, which corresponds to 45 degrees. This angle is known to have special and precise trigonometric values.

Step 2: Determine Sine and Cosine

For \( \theta = \frac{\pi}{4} \), the sine and cosine functions have equal values. Therefore, we have: \( \sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \). These are also often represented as \( \frac{\sqrt{2}}{2} \) after rationalizing the denominator.

Step 3: Calculate Tangent

The tangent of an angle is the ratio of the sine to the cosine. So for \( \theta = \frac{\pi}{4} \), we have:\[\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = 1.\]

Step 4: Determine Cosecant (Csc)

The cosecant is the reciprocal of the sine. Thus, for \( \theta = \frac{\pi}{4} \):\[\csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \sqrt{2}.\]

Step 5: Determine Secant (Sec)

The secant is the reciprocal of the cosine. Therefore, for \( \theta = \frac{\pi}{4} \):\[\sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \sqrt{2}.\]

Step 6: Determine Cotangent (Cot)

The cotangent is the reciprocal of the tangent. Hence, for \( \theta = \frac{\pi}{4} \):\[\cot\left(\frac{\pi}{4}\right) = \frac{1}{\tan\left(\frac{\pi}{4}\right)} = 1.\]

Key Concepts

Sine and Cosine

Sine and cosine are the most fundamental trigonometric functions. They are derived from the unit circle, which is a circle with a radius of 1. In the unit circle, for any angle \(\theta\), the x-coordinate represents the cosine value, while the y-coordinate represents the sine value. This makes understanding sine and cosine crucial for solving various trigonometric problems.
For \(\theta = \frac{\pi}{4}\), both sine and cosine have the same value. This is because at an angle of 45 degrees (or \(\frac{\pi}{4}\) radians), the coordinates on the unit circle are symmetrical, so:

• Sine: \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)
• Cosine: \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)

Understanding this symmetry is helpful, as it simplifies calculations and enhances comprehension of more complex trigonometric concepts.

Tangent and Cotangent

Tangent and cotangent relate the sine and cosine functions. The tangent of an angle is defined as the ratio of the sine to the cosine of that angle. For the angle \(\theta = \frac{\pi}{4}\), since sine and cosine are equal, the tangent simplifies to one:

• Tangent: \(\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = 1\)

The cotangent, on the other hand, is the reciprocal of the tangent, indicating the ratio of cosine to sine. Therefore, for \(\theta = \frac{\pi}{4}\), the cotangent is also one:

• Cotangent: \(\cot\left(\frac{\pi}{4}\right) = \frac{1}{\tan\left(\frac{\pi}{4}\right)} = 1\)

These definitions help enhance problem-solving skills in trigonometry by allowing transformation between different function forms.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include the cosecant, secant, and cotangent. These functions are the reciprocals of sine, cosine, and tangent, respectively. Understanding these functions is important because they often appear in trigonometric identities and equations.
For \(\theta = \frac{\pi}{4}\):

• Cosecant: \(\csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \sqrt{2}\)
• Secant: \(\sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \sqrt{2}\)
• Cotangent, as mentioned earlier, is 1 because it's the reciprocal of tangent.

By grasping these reciprocal relationships, you can tackle more complex trigonometric functions with ease.

Special Angles in Trigonometry

Special angles in trigonometry, like \(\frac{\pi}{4}\), \(\frac{\pi}{6}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), have precise trigonometric values that form the foundation for many trigonometric calculations. Memorizing the values of sine, cosine, and tangent for these angles helps in quickly solving trigonometry problems without a calculator.
These angles are derived from the symmetrical properties of the unit circle and the patterns recognized in right-angled triangles.

• For example, at \(\frac{\pi}{4}\): both \(\sin\left(\frac{\pi}{4}\right)\) and \(\cos\left(\frac{\pi}{4}\right)\) are \(\frac{1}{\sqrt{2}}\), as noted before.
• The tangent, as it is a ratio of sine to cosine, equals 1 at this angle.

Mastery of these special angles will greatly aid in simplifying and evaluating trigonometric expressions often used in calculus and other advanced mathematics topics.