MathematicsChapter 6: Trigonometry and Bearing
Angles and Standard Position
An angle is formed by two rays that share a common vertex.
Learning objectives
- Explain angles and standard position in clear mathematical language
- Apply the ideas in positive and negative
- Solve problems involving standard position
Definition
An angle is formed by two rays that share a common vertex. One ray is the initial arm and the other is the terminal arm. A positive angle is measured anti-clockwise from the initial arm. A negative angle is measured clockwise. In standard position the vertex sits at the origin and the initial arm lies along the positive x-axis. That placement lets you read an angle on the coordinate plane and later on the unit circle.
Where / symbols
- two rays
- An angle is formed by two rays that share a common vertex.
- common vertex
- An angle is formed by two rays that share a common vertex.
- positive
- A positive angle is measured anti-clockwise from the initial arm.
- anti-clockwise
- A positive angle is measured anti-clockwise from the initial arm.
Overview
Direction first, then the home position:
Positive and Negative
From the initial arm, swing anti-clockwise for a positive angle and clockwise for a negative angle. A turn of 90 deg anti-clockwise and a turn of -270 deg clockwise can end on the same terminal arm, but they are not the same directed angle. Sign tells you which way you rotated, not only where you stopped.
Examples +90 deg from the x-axis up to the y-axis · -90 deg from the x-axis down · +360 deg a full anti-clockwise turn
Standard Position
Put the vertex at the origin and the initial arm along the positive x-axis. The terminal arm then falls in a quadrant, or along an axis. An angle of 0 deg has its terminal arm on the positive x-axis. An angle of 90 deg has its terminal arm on the positive y-axis. Standard position is the agreed photo of an angle before you talk about sine and cosine.
Examples 0 deg along positive x-axis · 90 deg along positive y-axis · 180 deg along negative x-axis
What it means
An angle is formed by two rays that share a common vertex.
Main ideas
1. POSITIVE AND NEGATIVE: From the initial arm, swing anti-clockwise for a positive angle and clockwise for a negative angle. 2. STANDARD POSITION: Put the vertex at the origin and the initial arm along the positive x-axis.
Why it matters
Two rays and a common vertex make the angle; the plane around them lets you choose a direction. Always name the initial arm before you decide positive or negative. Standard position is not a special kind of angle; it is a special way of drawing one.
How to answer in an exam
Clockwise is negative in this course, even if a clock itself turns that way. Do not measure from the y-axis unless a question says so. Standard position starts from the positive x-axis at the origin.
Explanation
Two rays and a common vertex make the angle; the plane around them lets you choose a direction. Always name the initial arm before you decide positive or negative. Standard position is not a special kind of angle; it is a special way of drawing one.
In simple words
Clockwise is negative in this course, even if a clock itself turns that way. Do not measure from the y-axis unless a question says so. Standard position starts from the positive x-axis at the origin.
Key points
- two rays
- common vertex
- positive
- anti-clockwise
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angles and Standard Position
Sticky Notes · Quick Revision
Core Idea
- An angle is formed by two rays that share a common vertex.
Key Terms
- two rays: An angle is formed by two rays that share a common vertex.
- common vertex: An angle is formed by two rays that share a common vertex.
Main Pattern
- 1. POSITIVE AND NEGATIVE: From the initial arm, swing anti-clockwise for a positive angle and clockwise for a negative angle.
- 2. STANDARD POSITION: Put the vertex at the origin and the initial arm along the positive x-axis.
Worked Reminder
- +90 deg from the x-axis up to the y-axis
- -90 deg from the x-axis down
- +360 deg a full anti-clockwise turn
Common Mistake
- Clockwise is negative in this course, even if a clock itself turns that way. Do not measure from the y-axis unless a question says so. Standard position starts from the positive x-axis at the origin.
Exam Tip
- State the rule for angles and standard position, show each algebraic step, and check the final answer against the question.
60-second revision
- An angle is formed by two rays that share a common vertex.
- From the initial arm, swing anti-clockwise for a positive angle and clockwise for a negative angle.
- Put the vertex at the origin and the initial arm along the positive x-axis.
- Two rays and a common vertex make the angle; the plane around them lets you choose a direction.
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