Problem 32 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\sin \theta=0.8090\)

Step-by-Step Solution

Answer

54^0 and 126^0.

Step 1: Understanding the Range and Function

We need to find the angles  heta where  heta is between 0^0 and 360^0 such that  heta satisfies  ext{sin } heta = 0.8090. The sine function is positive in the first and second quadrants.

Step 2: Find First Quadrant Angle

Using inverse sine, find the angle in the first quadrant for  ext{sin } heta = 0.8090. Use a calculator to find  heta = ext{sin}^{-1}(0.8090) which gives approximately 54.06^0. Round this to  heta_1 = 54^0.

Step 3: Find Second Quadrant Angle

In the second quadrant, the sine of an angle  heta is equal to the sine of 180^0 - heta_1. Calculate this to find the second angle:  heta_2 = 180^0 - 54^0 = 126^0.

Step 4: Verify the Angles

Make sure both  heta_1 = 54^0 and  heta_2 = 126^0 fall within the given range of 0^0  heta < 360^0 and that they satisfy  ext{sin } heta = 0.8090 when rounded correctly.

Key Concepts

Inverse Sine Function

The inverse sine function is a powerful tool for finding angles when you know the sine value. When you see \( \sin^{-1}(x) \), it's asking, "What angle gives me this sine value?" Importantly, the result of the inverse sine function is always within its principal range, which is \(-90^{\circ}\) to \(90^{\circ}\) or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians. This range covers the first and fourth quadrants, which is where the function is defined.

• **Principal Range**: \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\)
• **Output**: Gives angle in first or fourth quadrant
• **Use**: Helps identify one possible solution of angles

For this exercise, when we used the inverse sine function on \( 0.8090 \), it yielded an angle of approximately \( 54^{\circ} \), which falls in the first quadrant. Once we have this angle, we have to remember that the sine function can repeat and may have additional solutions.

Quadrants in Trigonometry

Understanding the quadrants is crucial when dealing with trigonometric functions since they determine the sign of the values. The Cartesian plane is divided into four quadrants:

• 1st Quadrant: All trigonometric functions are positive.
• 2nd Quadrant: Sine is positive, cosine and tangent are negative.
• 3rd Quadrant: Tangent is positive, sine and cosine are negative.
• 4th Quadrant: Cosine is positive, sine and tangent are negative.

In this exercise, since we know that sine is positive for our angle \(\theta\), we understand that our solutions will lie in the first or second quadrants. For the given value \( \sin \theta = 0.8090 \), it confirms these quadrant placements, namely \( 54^{\circ} \) in the first quadrant and then applying symmetry \(126^{\circ}\) in the second quadrant.

Sine Function Properties

The sine function is periodic and symmetric, which means it repeats its values in regular intervals, and it behaves the same way in different quadrants. The most critical properties that come in handy are:

• **Periodicity**: Sine repeats every \(360^{\circ}\), or \(2\pi\) radians.
• **Symmetry**: Sine is an odd function, meaning \(\sin(-\theta) = -\sin(\theta)\).
• **Range and Extremes**: Sine values range from \(-1\) to \(1\), with maximum +1 occurring at \(90^{\circ}\) and minimum -1 at \(270^{\circ}\).

In our exercise, because of its periodic nature and symmetry about \(180^{\circ}\), once you find \(\theta_1 = 54^{\circ}\), another possible angle mirrored across \(180^{\circ}\) would be \(\theta_2 = 180^{\circ} - 54^{\circ} = 126^{\circ}\). This property allows for the multiplication of solutions inside a specific interval, such as \([0^{\circ}, 360^{\circ})\). Knowing these properties allows us to predict and calculate these multiple solutions efficiently.