Problem 27 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(\frac{3}{5}, \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

\(\sin t = \frac{4}{5}, \cos t = \frac{3}{5}, \tan t = \frac{4}{3}\).

Step 1: Identify Coordinates

The coordinates provided for the terminal point are \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \). These coordinates represent \( \cos t \) and \( \sin t \) respectively for the angle \( t \).

Step 2: Calculate Sinusoidal Values

Given that \( \cos t = \frac{3}{5} \) and \( \sin t = \frac{4}{5} \), we have already identified the values for \( \sin t \) and \( \cos t \). Thus, \( \sin t = \frac{4}{5} \) and \( \cos t = \frac{3}{5} \).

Step 3: Determine Tangent Value

The tangent of the angle, \( \tan t \), is calculated using the formula \( \tan t = \frac{\sin t}{\cos t} \). Substitute the values \( \sin t = \frac{4}{5} \) and \( \cos t = \frac{3}{5} \), so \( \tan t = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \).

Key Concepts

Understanding Sine

Sine is a fundamental trigonometric ratio that connects the angle in a right triangle to the lengths of the triangle's sides. Specifically, sine compares the length of the opposite side to the hypotenuse. If you visualize a right triangle within a circle, the sine of the angle corresponds to the y-coordinate of the point on the circle's circumference. Thus, the terminal point coordinates help us easily identify the sine value.

If you consider an angle, say \( t \), the sine of that angle, noted as \( \sin t \), is determined by \( y \) in the point \((x, y)\). For example, given the terminal point \( \left(\frac{3}{5}, \frac{4}{5}\right) \), the sine of \( t \) is simply the \( y \)-coordinate, which is \( \frac{4}{5} \).

• Always remember: Sine = Opposite / Hypotenuse
• In a unit circle, where the radius = 1, sine is the y-coordinate of the point.

Exploring Cosine

Cosine is another vital trigonometric ratio closely related to sine. The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. Just like sine, if you think of a circle, the cosine relates to the x-coordinate of a terminal point on the unit circle.

Given an angle \( t \), the cosine, written as \( \cos t \), can quickly be identified as the \( x \)-coordinate of the point \((x, y)\). For instance, with a point like \( \left(\frac{3}{5}, \frac{4}{5}\right)\), the value of \( \cos t \) is the \( x \)-coordinate, which in this case, is \( \frac{3}{5} \).

• Remember: Cosine = Adjacent / Hypotenuse
• On the unit circle, cosine is represented by the x-coordinate.

Interpreting Tangent

Tangent is a crucial trigonometric concept that links both sine and cosine together. The tangent of an angle is the ratio of sine to cosine. Visually, when considering the unit circle, tangent can also be thought of as the slope of the line formed by the radius at a given angle.

To find the tangent of an angle \( t \), known as \( \tan t \), you simply divide \( \sin t \) by \( \cos t \). For our coordinates, \( \tan t \) becomes \( \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \).

• Key formula: Tangent = Opposite / Adjacent = Sine / Cosine
• In the context of the unit circle, remember that tangent represents the slope of the line from the origin to the terminal point.