Problem 94 - Precalculus : Building Concepts and Connections

Question

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$(0.8,-0.6)$$

Step-by-Step Solution

Answer

The cosine of the angle \(z\) is 0.8 and the sine of the angle \(z\) is -0.6.

Step 1: Identifying the Coordinates on Unit Circle

Identify the given point which is at (0.8, -0.6). The coordinates represent the value of cosine and sine respectively.

Step 2: Calculating Cosine

Calculate the cosine of the angle, which is the \(x\)-coordinate of the given point. Thus, the cosine of the angle \(z\) is 0.8.

Step 3: Calculating Sine

Calculate the sine of the angle, which is the \(y\)-coordinate of the given point. Thus, the sine of the angle \(z\) is -0.6.

Key Concepts

Sine and Cosine

The sine and cosine are fundamental functions in trigonometry, often represented as sin and cos. They are defined based on the unit circle, which is a circle of radius one centered at the origin of a coordinate system. Picture the unit circle in your mind, or better yet, draw one.

In the context of our right-angled triangle, sine and cosine relate to the sides of the triangle. The sine of an angle is the length of the side opposite the angle divided by the hypotenuse, which in the unit circle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. Conversely, cosine relates to the length of the adjacent side to the angle over the hypotenuse, corresponding to the x-coordinate on the unit circle.

When we're given a point like (0.8, -0.6), we are directly provided with the values for cosine and sine for the angle that reaches that point on the unit circle: cosine is 0.8, and sine is -0.6.

Standard Position Angle

An angle is said to be in standard position when its vertex is at the origin and it starts from the positive x-axis, extending to the terminal side in a counterclockwise or clockwise direction. Counterclockwise usually indicates a positive angle, while a clockwise direction indicates a negative angle.

Finding Standard Position

A technique to find the standard position of an angle is to visualize rotating a ray from the positive x-axis through the given angle, with the point where the ray intersects the unit circle marking the terminal side. The terminal point provides the trigonometric coordinates which represent the sine and cosine of the angle. In our case, the angle ends at the point (0.8, -0.6), so its standard position has a terminal side that lies below the x-axis because the y-coordinate is negative.

Determining the exact angle in standard position requires inverse trigonometric functions or further contextual information.

Trigonometric Coordinates

Trigonometric coordinates, often associated with a point on the unit circle, represent the cosine and sine of an angle whose terminal side passes through that point. It's important to note these coordinates will always satisfy the relation x^2 + y^2 = 1, because of the Pythagorean Theorem which states that in a right-angled triangle, the sum of squares of the legs is equal to the square of the hypotenuse.

The Connection to Angles

Every point (x, y) on the unit circle is associated with an angle in standard position. For instance, the point (0.8, -0.6) is connected with our angle z, and from this, we can derive that the cosine of z is 0.8 (the x-value) and the sine of z is -0.6 (the y-value). Remember, this correlation is only valid for angles in standard position intersecting the unit circle.