91Ó°ÊÓ

The sine function is one of the primary trigonometric functions and is often abbreviated as \( \sin \). It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
The value of the sine function oscillates between -1 and 1 as the angle \( \theta \) varies within the interval \( [0^{\circ}, 360^{\circ}) \).

• In the first quadrant, \( \sin(\theta) \) is positive.
• Similarly, in the second quadrant, \( \sin(\theta) \) is also positive.
• However, in the third and fourth quadrants, \( \sin(\theta) \) becomes negative.

When solving the equation \( \sin \theta = 0.8225 \), we find the principal angle using a calculator or a sine table, giving us \( \theta = 55.0^{\circ} \).
Since sine is positive in both the first and second quadrants, we also consider the angle \( 180^{\circ} - 55.0^{\circ} = 125.0^{\circ} \).
Thus, the solutions are \( \theta = 55.0^{\circ} \) and \( \theta = 125.0^{\circ} \).

The cosine function, denoted as \( \cos \), is another fundamental trigonometric function. It represents the ratio of the adjacent side to the hypotenuse in a right-angle triangle.
The cosine function ranges from -1 to 1 as the angle moves through various quadrants.

• In the first quadrant, \( \cos(\theta) \) is positive.
• It's negative in the second quadrant and remains negative in the third quadrant.
• In the fourth quadrant, \( \cos(\theta) \) becomes positive again.

When addressing the equation \( \cos \theta = -0.6604 \), we utilize the inverse cosine function to find \( \theta = 131.5^{\circ} \).
As the cosine is negative in the second and third quadrants, we deduce another angle as \( 360^{\circ} - 131.5^{\circ} = 228.5^{\circ} \).
Therefore, the answers are \( \theta = 131.5^{\circ} \) and \( \theta = 228.5^{\circ} \).

The tangent function, symbolized by \( \tan \), is defined as the ratio of the sine to the cosine of an angle. In a right triangle, it represents the ratio of the opposite side to the adjacent side.
Unlike the sine and cosine functions, the tangent function does not have a maximum or minimum value but rather extends from negative to positive infinity within its periodic intervals.

• Tangent is positive in the first quadrant and negative in the second.
• It's positive again in the third quadrant and negative in the fourth.

For the function \( \tan \theta = -1.5214 \), using the inverse tangent function results in approximately \( \theta = -56.4^{\circ} \).
Adjusting it to the given interval, \( \theta = 360^{\circ} - 56.4^{\circ} = 303.6^{\circ} \).
Since tangent is negative in the second quadrant, another angle is \( 180^{\circ} - 56.4^{\circ} = 123.6^{\circ} \).
The angles that satisfy are \( 123.6^{\circ} \) and \( 303.6^{\circ} \).

The cotangent function, denoted as \( \cot \), is the reciprocal of the tangent function. It equates to the ratio of the cosine of the angle to its sine.

• In right-angled triangles, it pairs the lengths of the adjacent side to the opposite side.
• The cotangent curve shares some resemblance with the tangent but reflects the inverse relationship.

To determine \( \theta \) from \( \cot \theta = 1.3752 \), first convert to the tangent by finding its reciprocal, yielding \( \tan \theta \approx 0.7273 \).
Using the inverse tangent to find \( \theta \), we obtain about \( 36.0^{\circ} \).
As tangent is positive in the first and third quadrants, the additional angle is \( 180^{\circ} + 36.0^{\circ} = 216.0^{\circ} \).
Thus, the solutions are \( 36.0^{\circ} \) and \( 216.0^{\circ} \).

The secant function, denoted by \( \sec \), is the reciprocal of the cosine function.
It measures the ratio of the hypotenuse to the adjacent side in a right triangle.
Like the tangent and cotangent functions, secant doesn’t have a specific range but has vertical asymptotes where cosine equals zero.

• Secant is positive in the first and fourth quadrants.
• It becomes negative in the second and third quadrants due to the cosine values.

Examining \( \sec \theta = 1.4291 \), take its reciprocal to find \( \cos \theta \approx 0.6999 \).
Using the inverse cosine function, we get \( \theta = 45.5^{\circ} \).
Since cosine remains positive in the first and fourth quadrants, another angle would be \( 360^{\circ} - 45.5^{\circ} = 314.5^{\circ} \).
Thus, \( \theta \) can be \( 45.5^{\circ} \) or \( 314.5^{\circ} \).

The cosecant function, represented as \( \csc \), is the reciprocal of the sine function.
It calculates the ratio of the hypotenuse to the opposite side in a triangle.
Like the secant, the cosecant function has no maximum or minimum value, possessing asymptotes where sine equals zero.

• Cosecant is positive in the first and second quadrants because it mirrors the sine.
• Thus, it turns negative in the third and fourth quadrants.

Solving \( \csc \theta = -2.3179 \), we determine \( \sin \theta \approx -0.4315 \) using its reciprocal.
Applying the inverse sine equation yields \( \theta \approx -25.6^{\circ} \).
Adjust to the interval by taking \( 360^{\circ} - 25.6^{\circ} = 334.4^{\circ} \).
Since sine is negative in the third quadrant, another solution is \( 180^{\circ} + 25.6^{\circ} = 205.6^{\circ} \).
Thus, \( \theta \) is \( 205.6^{\circ} \) and \( 334.4^{\circ} \).