Question

$$f(x)=\sin x, g(x)=\cos x, h(x)=\tan x, F(x)=\csc x, G(x)=\sec x, \text{ and } H(x)=\cot x$$ (a) Find \(f\left(\frac{7 \pi}{4}\right) .\) What point is on the graph of \(f ?\) (b) Find \(G\left(\frac{7 \pi}{4}\right) .\) What point is on the graph of \(G ?\) (c) Find \(F\left(-225^{\circ}\right)\). What point is on the graph of \(F ?\)

Short answer

Values are: \(f\left(\frac{7 \pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \(G\left(\frac{7 \pi}{4}\right) = \sqrt{2}\), \(F\left(-\frac{5 \pi}{4}\right) = -\sqrt{2}\).

Step-by-step solution

- Convert radians to degrees if needed

For parts (a) and (b), the input is already in radians. For part (c), convert \(-225^{\circ}\) to radians using the formula: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\).

- Find \(f\left(\frac{7 \pi}{4}\right)\)

Method: \(f(x) = \sin x\).Put \(x = \frac{7 \pi}{4}\):\[ \sin \left(\frac{7 \pi}{4}\right) = -\frac{\sqrt{2}}{2}\] Therefore, the point on the graph is \(\left(\frac{7 \pi}{4}, -\frac{\sqrt{2}}{2}\right)\).

- Find \(G\left(\frac{7 \pi}{4}\right)\)

Method: \(G(x) = \sec x = \frac{1}{\cos x}\).Put \(x = \frac{7 \pi}{4}\):\[ \sec \left(\frac{7 \pi}{4}\right) = \frac{1}{\cos\left(\frac{7 \pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\] Therefore, the point on the graph is \(\left(\frac{7 \pi}{4}, \sqrt{2}\right)\).

- Convert degrees to radians for \(F(-225^{\circ})\)

Method: Multiply the degree value by \(\frac{\pi}{180}\). \(-225^{\circ} \times \frac{\pi}{180} = -\frac{5\pi}{4}\).

- Find \(F\left(-\frac{5 \pi}{4}\right)\)

Method: \(F(x) = \csc x = \frac{1}{\sin x}\). Put \(x = -\frac{5 \pi}{4}\): \[ \csc \left(-\frac{5 \pi}{4}\right) = \frac{1}{\sin\left(-\frac{5 \pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\] Therefore, the point on the graph is \(\left(-\frac{5 \pi}{4}, -\sqrt{2}\right)\).

Key concepts

Sine Function

The sine function, denoted as \(\text{sin}(x)\), is one of the primary trigonometric functions. It describes the ratio of the length of the opposite side to the hypotenuse in a right triangle. When dealing with the unit circle, \(\text{sin}(x)\) represents the y-coordinate of a point on the circle. For example, in the given exercise, we found \(\text{sin} \frac{7\text{π}}{4} = -\frac{\text{√}2}{2} \). This result is achieved by finding the position of \(\frac{7\text{π}}{4}\) on the unit circle and determining the corresponding y-coordinate.
Remember, trigonometric functions are periodic. For sine, the period is \2\text{π}\, meaning \(\text{sin}(x + 2\text{π}) = \text{sin}(x)\).

Secant Function

The secant function, represented as \(\text{sec}(x)\), is the reciprocal of the cosine function. It can be written as \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\). When we are finding \(\text{sec} \frac{7\text{π}}{4}\), we're actually determining \(\frac{1}{\text{cos} \frac{7\text{π}}{4}}\). Given that \(\text{cos} \frac{7\text{π}}{4} = \frac{\text{√}2}{2}\), we can conclude that \(\text{sec} \frac{7\text{π}}{4} = \frac{1}{\frac{\text{√}2}{2}} = \text{√}2\).
This function is also periodic with a period of \2\text{π}\, like the cosine function.

Cosecant Function

The cosecant function, \(\text{csc}(x)\), is the reciprocal of the sine function, expressed as \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\). To find \(\text{csc}(-225^{\text{°}})\), we first convert the angle from degrees to radians which results in \(-225^{\text{°}} = -\frac{5\text{π}}{4}\). Then, we evaluate \(\text{sin}(-\frac{5\text{π}}{4}) = -\frac{\text{√}2}{2}\), so \(\text{csc}(-\frac{5\text{π}}{4}) = \frac{1}{-\frac{\text{√}2}{2}} = -\text{√}2\).
This function also has a period of \2\text{π}\.

Radian to Degree Conversion

Converting between radians and degrees is crucial in trigonometry, as different problems might provide angles in either unit. The formula for converting degrees to radians is \( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\text{π}}{180} \). For instance, to convert \-225^{\text{°}} \ to radians, we use \(-225 \times \frac{\text{π}}{180} = -\frac{5\text{π}}{4} \).
Conversely, to convert radians to degrees, use \( \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\text{π}} \).