Find R and Theta

Find R and Theta

What is it?

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

In simple words

• r = sqrt(x²+y²).
• 1+i: r=sqrt(2), theta=pi/4
• -1: r=1, theta=pi
• State the rule, show the working, and check the answer.

Important terms

FIND R AND THETA: r = sqrt(x^2+y^2). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

FIND r AND theta: r = sqrt(x²+y²).

What you should learn

• State the rule or meaning of find r and theta in easy words.
• Apply find r and theta in a clear step-by-step solution.
• Check the conditions, working, and final answer for accuracy.

Examples

• 1+i: r=sqrt(2), theta=pi/4
• -1: r=1, theta=pi
• i: r=1, theta=pi/2

MAIN IDEA

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

EXAM USE

Write the rule first: r = sqrt(x^2+y^2). Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

Full school-level explanation

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

Understand the idea, then apply it:

WHAT — FIND R AND THETA

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

WHY — Why it matters

This rule is a separate learning unit inside argand plane and polar form. It can be defined, applied, and checked in its own FBISE question.

HOW — How to answer

Write the rule first: r = sqrt(x^2+y^2). Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

MAIN IDEA

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

Remember

• quadrant first

Examples

• 1+i: r=sqrt(2), theta=pi/4
• -1: r=1, theta=pi
• i: r=1, theta=pi/2

EXAM USE

Write the rule first: r = sqrt(x^2+y^2). Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

Remember

• focused answer
• accurate support

Examples

• State the central idea
• Use one accurate detail
• Check the final response

How to answer a find r and theta question

How it works

• Read the exact command word and identify what the question asks.
• State the matching idea, rule, or meaning before adding detail.
• Use the unit evidence or mathematical working that directly supports the answer.
• Check accuracy, relevance, and answer length before finishing.

Remember

• quadrant first
• r = |z|
• z = r (cos theta + i sin theta)
• quadrant
• quadrant first
• focused answer
• accurate support
• quadrant first

Deeper understanding

Go beyond memorising Find R and Theta. Connect the definition to reasons, relationships, exceptions, and exam wording.

Common mistakes

Exam focus

Core Idea

What to Know

Example or Evidence

How to Answer

Exam focus

Quick revision

• quadrant first
• r = |z|
• z = r (cos theta + i sin theta)
• quadrant
• FIND R AND THETA: r = sqrt(x^2+y^2). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly
• FIND r AND theta: r = sqrt(x²+y²).

theta must match the quadrant. (-1)+i is not pi/4.

For long answers on Find R and Theta: define clearly → explain the main parts → give one example → state one exam caution.