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Chapter 1: Problem 66

Find the supplement of each of the following angles. $$ 30^{\circ} $$

Short Answer

Expert verified
The supplement of 30掳 is 150掳.

Step by step solution

01

Understanding the Problem

To find the supplement of an angle, we need to determine the angle that, when added to the given angle, equals 180掳.
02

Applying the Supplementary Angle Formula

The formula to determine the supplementary angle is: \[ \text{Supplementary angle} = 180^{\circ} - \text{given angle} \]
03

Substitute the Given Value

Substitute the given angle into the formula: \[ \text{Supplement} = 180^{\circ} - 30^{\circ} \]
04

Calculate the Supplement

Perform the subtraction: \[ \text{Supplement} = 150^{\circ} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angle subtraction
Understanding angle subtraction is key when dealing with supplementary angles. The term "angle subtraction" refers to the process of finding the difference between 180掳 and a given angle, which allows us to determine its supplement. The main aim is to determine what angle, when added to the given one, makes 180掳. To make it simple:
  • Identify the given angle.
  • Subtract this angle from 180掳 using the formula.
When working with the angle 30掳, we subtract it from 180掳 to find the supplement, concluding that the result is 150掳. Remember, subtraction in this context is merely a tool to balance the scale of an angle to reach the full line, turning into a straight angle which is always 180掳.
trigonometry concepts
Trigonometry often deals with the relationships between angles and distances in geometry. However, it鈥檚 not just for triangles and circles. Here, in the case of supplementary angles, we use simple arithmetic tools from trigonometry:
  • Supplementary angles.
  • Fundamental angle relationships derived from these.
When dealing with simple angle pairs, such as 30掳 and its supplement 150掳, remember that they are part of an imaginary straight line. This line represents 180掳, and any pair of angles lying on it are mutually supplementary. Understanding these trigonometry basics greatly helps when approaching more complex topics involving relationships and calculations of various angles.
angle relationships
Angle relationships can serve as a fascinating insight into geometry basics. Understanding them simplifies how we see functions of angles in various shapes and forms. Complementary and supplementary relationships are vital:
  • Supplementary angles always add up to 180掳.
  • Complementary angles total 90掳.
With supplementary pairs, like our 30掳 angle example, the missing piece to complete the straight line is easy to identify through basic subtraction. Keep these relationships in mind, as many geometric problems boil down to recognizing and working with these fundamental angle relationships. Learning to navigate through them gives you a strong geometry foundation and aids in recognizing patterns across different math problems.

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Most popular questions from this chapter

For Problems 23 through 26 , recall that \(\sin ^{2} \theta\) means \((\sin \theta)^{2}\). If \(\tan \theta=2\), find \(\tan ^{3} \theta\)

Use the reciprocal identities for the following problems. If \(\tan \theta=a(a \neq 0)\), find \(\cot \theta\)

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \sin \theta(\csc \theta-\sin \theta)=\cos ^{2} \theta $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \frac{\sec \theta}{\tan \theta}=\csc \theta $$

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\cos \theta\) if \(\sin \theta=-\frac{1}{4}\) and \(\theta\) terminates in QII.
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