Trig Review: Videos & Practice Problems

Understanding vectors involves recognizing their components across all quadrants of the x-y plane. Magnitudes are always positive, while components can be positive or negative based on direction. Reference angles are crucial for calculations, particularly when using equations like $A=A\cos \left(\theta \right)$ and $A=A\sin \left(\theta \right)$. Additionally, angles can be described in terms of clockwise or counterclockwise directions, affecting their sign in calculations. Recognizing cardinal directions like "30 degrees north of east" aids in vector representation.

Doing Math with Vectors in any Quadrant (More Trig)

Understanding vectors in the Cartesian coordinate system requires familiarity with their components and how they behave in different quadrants. Regardless of the quadrant, the magnitudes of vectors are always positive. However, the components of these vectors can be positive or negative based on their direction. In the first quadrant, both components are positive, while in the second quadrant, the x-component is negative and the y-component is positive. In the third quadrant, both components are negative, and in the fourth quadrant, the x-component is positive and the y-component is negative. This distinction is crucial for accurately representing vectors in any quadrant.

To break down a vector into its components, we often use trigonometric functions. The x-component can be calculated using the cosine function, while the y-component is derived from the sine function. When given an angle relative to the y-axis, it is essential to convert it to the reference angle relative to the x-axis. This can be done using the relationship:

$$\theta_x = 90^\circ - \theta_y$$

where $$\theta_y$$ is the angle measured from the y-axis. This conversion allows for the correct application of the trigonometric functions to find the components.

In addition to reference angles, understanding absolute angles is important. The absolute angle is measured from the positive x-axis and can be calculated by considering the reference angle and the quadrant in which the vector lies. For example, in the second quadrant, the absolute angle can be found using:

$$\theta_{absolute} = 180^\circ + \theta_{reference}$$

In practice, to find the components of a vector given its magnitude and angle, one would first determine the reference angle and then apply the cosine and sine functions accordingly. For instance, if a vector has a magnitude of 13 and an angle of 22.6 degrees relative to the y-axis, the reference angle would be:

$$\theta_x = 90^\circ - 22.6^\circ = 67.4^\circ$$

Then, the components can be calculated as:

$$A_x = A \cdot \cos(\theta_x)$$

$$A_y = A \cdot \sin(\theta_x)$$

In this case, the x-component would be negative (pointing left) and the y-component would be positive (pointing up). Finally, to find the absolute angle, one would add the reference angle to 180 degrees:

$$\theta_{absolute} = 180^\circ + 67.4^\circ = 247.4^\circ$$

By mastering these concepts, students can effectively analyze and solve problems involving vectors in any quadrant of the Cartesian plane.

Vector F is 65 m long, directed 30.5° below the positive x-axis. (a) Find the x-component, F_x. (b) Find the y-component, F_y.

• A. F_x = 30.6 m, F_y = 51.6 m
• B. F_x = 56 m, F_y = 33 m
• C. F_x = 30.6 m, F_y = − 51.6 m
• D. F_x = 56 m, F_y = − 33 m

Answer: (D)

The vector A represented is by the pair of components A_x = -77 cm, A_y = 36 cm. (a) Find the magnitude of vector A. (b) Find the absolute angle of this vector.

• A. (a) A = 68 cm (b) 179.6°
• B. (a) A = 85 cm (b) 179.6°
• C. (a) A = 68 cm (b) 154.9°
• D. (a) A = 85 cm (b) 154.9°
• E. (a) A = 68 cm (b) −25.1°
• F. (a) A = 85 cm (b) −25.1°

Answer: (D)

Describing Vectors with Words (More Trig)

When solving vector problems, understanding the terminology used to describe directions is crucial. Key terms include counterclockwise (CCW) and clockwise (CW), as well as cardinal directions like northeast and southwest. In this context, counterclockwise angles are considered positive, while clockwise angles are negative. For instance, a 45-degree angle measured counterclockwise from the x-axis is simply +45°, whereas a 60-degree angle measured clockwise is represented as -60°.

Despite the positive or negative sign indicating the direction of the angle, the reference angle used in component equations, such as cosine and sine, is always a positive value relative to the nearest x-axis. For example, if you have a vector of 5 meters at +37° from the negative x-axis, you would calculate the components using:

$$A_x = A \cdot \cos(37°)$$

$$A_y = A \cdot \sin(37°)$$

Here, the x-component would be 5 * cos(37°) = 4, and the y-component would be 5 * sin(37°) = 3. Since this vector lies in the third quadrant, both components are negative.

For a vector with a clockwise angle from the positive y-axis, such as 53 degrees, the reference angle is determined by its complementary angle. Thus, you would still use the positive reference angle of 37° for calculations:

$$B_x = B \cdot \cos(37°)$$

$$B_y = B \cdot \sin(37°)$$

In this case, both components would be positive since they point to the right and upward in the first quadrant.

Additionally, vectors can be described using cardinal directions. For example, a vector of 6 units at 30 degrees north of east involves drawing the vector starting from the east direction and curving towards the north. The x-component is calculated as:

$$A_x = 6 \cdot \cos(30°)$$

resulting in approximately 5.2. Conversely, for a vector of 10 units at 53 degrees west of south, you would first draw the vector south and then curve towards the west. The reference angle for calculations would be 37 degrees, leading to:

$$B_x = 10 \cdot \cos(37°)$$

yielding a positive value, but conventionally, the westward direction is considered negative, so the final x-component would be -8.

Understanding these directional conventions and how to calculate vector components is essential for accurately solving vector problems in physics and engineering.

A small helicopter travels 225 m across a city in a direction 53.1° south of east. What are the components of the helicopter's trip?

• A. 135 m east (+x), 180 m south (-y)
• B. 180 m east (+x), 135 m south +y)
• C. 214 m east (-x), 68 m south (-y)
• D. 68 m east (+x), 214 m south (+y)

Answer: (A)

Here’s what students ask on this topic:

What are the signs of vector components in different quadrants?

The signs of vector components depend on the quadrant in which the vector lies. In Quadrant I, both x and y components are positive. In Quadrant II, the x component is negative, and the y component is positive. In Quadrant III, both x and y components are negative. In Quadrant IV, the x component is positive, and the y component is negative. This is because positive components point up and to the right, while negative components point down and to the left.

How do you find the reference angle for a vector?

To find the reference angle for a vector, you need the angle relative to the x-axis. If given an angle relative to the y-axis (θ_y), subtract it from 90° to get the reference angle (θ_x). For example, if θ_y is 10°, then θ_x = 90° - 10° = 80°. This reference angle is used in equations like A = A cos(θ) and A = A sin(θ).

What is the difference between clockwise and counterclockwise angles?

Clockwise angles are measured in the direction of the clock's hands and are considered negative. Counterclockwise angles are measured against the clock's hands and are considered positive. For example, a 45° counterclockwise angle is +45°, while a 60° clockwise angle is -60°. However, when using these angles in component equations, the reference angle is always positive.

How do you calculate the absolute angle of a vector?

The absolute angle of a vector is the angle measured from the positive x-axis to the vector. To find it, you may need to add or subtract known angles. For example, if the reference angle is 37° in the second quadrant, the absolute angle is 180° - 37° = 143°. This angle is different from the reference angle, which is relative to the nearest x-axis.

How do you interpret directions like '30 degrees north of east'?

To interpret '30 degrees north of east,' start by drawing an arrow pointing east. Then, curve the arrow 30° towards the north. This means the vector is 30° above the east direction. For calculations, use the reference angle relative to the x-axis, which in this case is 30°.