교과서 질문
In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
5장, 문제 33
In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)
검증된 단계별 안내1
Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), which is given by the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
Identify the given polar coordinates: \(r = 4\) and \(\theta = 90^\circ\).
Convert the angle \(\theta\) from degrees to radians if necessary, but since trigonometric functions can be evaluated directly in degrees, you can use \(90^\circ\) as is.
Calculate the rectangular coordinates using the formulas: \(x = 4 \cos(90^\circ)\) and \(y = 4 \sin(90^\circ)\).
Evaluate the cosine and sine values for \(90^\circ\) and substitute them back into the expressions for \(x\) and \(y\) to find the rectangular coordinates.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Polar Coordinates
Polar coordinates represent a point in the plane using a distance from the origin (radius) and an angle measured from the positive x-axis. The format is (r, θ), where r is the radius and θ is the angle in degrees or radians.
추천 영상:
05:32
Intro to Polar Coordinates
Conversion from Polar to Rectangular Coordinates
To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the formulas x = r * cos(θ) and y = r * sin(θ). This translates the point from a radius and angle to Cartesian x and y values.
추천 영상:
06:17Convert Points from Polar to Rectangular
Trigonometric Functions and Angle Measurement
Understanding sine and cosine functions is essential, as they relate angles to ratios of sides in right triangles. Also, knowing how to work with angles in degrees or radians ensures correct evaluation of these functions during conversion.
추천 영상:
6:04Introduction to Trigonometric Functions
관련 실천
교과서 질문
In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3
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교과서 질문
In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 20(cos 205° + i sin 205°)
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교과서 질문
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π
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In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of 8i
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