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Chapter 1: Problem 7

Find all angles \(0^{\circ} \leq \theta<360^{\circ}\) which satisfy the given equation: $$\sin \theta=0.4226$$

Short Answer

Expert verified
The angles are \(\theta = 25^{\circ}, 155^{\circ}\).

Step by step solution

01

- Determine the reference angle

Find the reference angle by taking the inverse sine of 0.4226. Use a calculator to find \(\theta_{ref} = \sin^{-1}(0.4226) \approx 25^{\circ}\).
02

- Identify Quadrants

Determine in which quadrants sine is positive. Since \(\sin \theta \) is positive in the first and second quadrants, we will use these to find the solutions.
03

- Calculate the angle in the first quadrant

In the first quadrant, the angle \(\theta\) is equal to the reference angle: \(\theta_{1} = 25^{\circ}\).
04

- Calculate the angle in the second quadrant

In the second quadrant, the angle \(\theta\) can be found using \(\theta_{2} = 180^{\circ} - 25^{\circ} = 155^{\circ}\).
05

- List all solutions

Combine the angles from the first and second quadrants for the final solution set: \(\theta = 25^{\circ}, 155^{\circ}\).

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Key Concepts

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

Inverse Trigonometric Functions
Inverse trigonometric functions help us find angles when given trigonometric values. In our exercise, we need to solve the equation \(\sin \theta = 0.4226\). To start, we use the inverse sine function, denoted as \(\sin^{-1}\) or \
Reference Angle
The reference angle is the acute angle a trigonometric function makes with the x-axis. In the first quadrant, the reference angle is the same as the given angle. Here, \(\theta_{\text{ref}} = 25掳\). For angles in other quadrants, you use different rules based on the function. In our case, we use it to find corresponding angles in quadrants where sine is positive.
Sine Function
The sine function relates to the y-coordinate of a point on the unit circle. It鈥檚 periodic with a period of 360掳 (or 2蟺 radians). When given \(\sin \theta = 0.4226\), we need to find all angles where the sine value is 0.4226. Since \(\sin \theta\) is positive in the first and second quadrants, we use this to determine relevant angles.
Quadrants in Trigonometry
Understanding quadrants helps determine where trigonometric values are positive or negative. The unit circle is divided into four quadrants:
  • First Quadrant: 0掳 to 90掳 (all trigonometric functions are positive).
  • Second Quadrant: 90掳 to 180掳 (sine is positive).
  • Third Quadrant: 180掳 to 270掳 (tangent is positive).
  • Fourth Quadrant: 270掳 to 360掳 (cosine is positive).
For our exercise, sine is positive in the first (25掳) and second quadrants (155掳 = 180掳 - 25掳).

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Most popular questions from this chapter

If the lengths \(a, b\), and \(c\) of the sides of a right triangle are positive integers, with \(a^{2}+b^{2}=c^{2}\), then they form what is called a Pythagorean triple. The triple is normally written as \((a, b, c) .\) For example, \((3,4,5)\) and \((5,12,13)\) are well-known Pythagorean triples. (a) Show that \((6,8,10)\) is a Pythagorean triple. (b) Show that if \((a, b, c)\) is a Pythagorean triple then so is \((k a, k b, k c)\) for any integer \(k>0\). How would you interpret this geometrically? (c) Show that \(\left(2 m n, m^{2}-n^{2}, m^{2}+n^{2}\right)\) is a Pythagorean triple for all integers \(m>n>0\). (d) The triple in part(c) is known as Euclid's formula for generating Pythagorean triples. Write down the first ten Pythagorean triples generated by this formula, i.e. use: \(m=2\) and \(n=1 ; m=3\) and \(n=1,2 ; m=4\) and \(n=1,2,3 ; m=5\) and \(n=1\), \(2,3,4\)

Let \(\theta=32^{\circ}\). Find the angle between \(0^{\circ}\) and \(360^{\circ}\) which is the (a) reflection of \(\theta\) around the \(x\) -axis (b) reflection of \(\theta\) around the \(y\) -axis (c) reflection of \(\theta\) around the origin

Find the values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. $$\sin A=\frac{3}{4}$$

Find all angles \(0^{\circ} \leq \theta<360^{\circ}\) which satisfy the given equation: $$\sin \theta=-0.6294$$

Find the values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. $$\tan A=\frac{5}{9}$$
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