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Calculus: Early Transcendentals

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

The image illustrates an angle with a radian measure of θ, centered at the vertex of a circle with a radius of 2.7 cm. The diagram shows the circle and the terminal side of the angle ending at a point on the circle.

**Diagram Details:**

- The circle is drawn with a blue outline and has a radius of 2.7 cm.
- The angle is positioned such that it intersects the positive x-axis.
- A right triangle is formed with the circle's radius, the x-axis, and a vertical line (from the x-axis to the terminal point of the angle).
- Measurements provided:
- The horizontal leg (adjacent to angle θ) measures 1.68 cm.
- The vertical leg (opposite angle θ) measures 2.11 cm.
- The hypotenuse of the triangle, which is the radius of the circle, measures 2.7 cm.

**Problem Statement:**

1. **Horizontal Distance Analysis:**
- The terminal point's horizontal distance to the right of the center of the circle is compared to the circle's radius.
- Formula: \(\cos(\theta) = \frac{\text{horizontal distance}}{\text{radius}}\)

2. **Vertical Distance Analysis:**
- The terminal point's vertical distance above the center of the circle is compared to the circle's radius.
- Formula: \(\sin(\theta) = \frac{\text{vertical distance}}{\text{radius}}\)

**Interactive Components:**

- Text boxes are provided to calculate and enter the values of \(\cos(\theta)\) and \(\sin(\theta)\).
- A "Preview" button likely allows users to check their answers.

These activities help in understanding how the trigonometric functions cosine and sine relate to a circle's geometry and facilitate practical application of the unit circle concept.

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A skier skis along a circular ski trail that has a radius of 1.3 km. The skiier starts at the East side of the ski trail and travels in the CCW direction. Let 0 represent the radian measure of the angle the skier has swept out. Suppose cos(0) = 0.85 and sin(0) = 0.52. %3D a. What does the 0.85 in cos(0) 0.85 represent in this context? Select all that apply. %3D O The skiier is 0.85 radius lengths to the East of the center of the ski trail. O The skiier is 0.52 km to the North of the center of the ski trail. O The skiier is 0.52 radius lengths to the North of the center of the ski trail. The skiier is 0.85 km to the East of the center of the ski trail. b. What does the 0.52 in sin(0) = 0.52 represent in this context? Select all that apply. %3D O The skiier is 0.85 radius lengths to the East of the center of the ski trail. The skiier is 0.52 radius lengths to the North of the center of the ski trail. The skiier is 0.52 km to the North of the center of the ski trail. O The skiier is 0.85…
b. Diagram 3 shows sector OPQ with centre O and sector PXY with centre P. Diagram 3 Given that OQ = 9 cm, PY = 4 cm, z XPY = 2.2 radians and the length of arc PQ = 7cm, calculate the value of 0, in radian. the area, in cm?, of the shaded region. i. ii.
If sin(θ)=1/5 and (θ) is in the 1st quadrant, find cos(θ). cos(θ)=? If cos(θ)=5/7 and θ is in the 4th quadrant, find the exact value for sin(θ).sin(θ)=? The point P is on the unit circle. If the y-coordinate of P is −4/5, and P is in quadrant IV , then x=?? If θ= −7π/6, then (in exact values, no deicimals) sin(θ)=?? cos(θ)=?? If θ= −1π/3, then (in exact values, no deicimals) sin(θ)=?? cos(θ)=?? If θ= 1π/4, then (in exact values, no deicimals) sin(θ)=?? cos(θ)=??

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