Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as sin⁻¹ (arcsin) and cos⁻¹ (arccos), are used to find angles when the value of a sine or cosine is known. For example, sin⁻¹(x) gives the angle whose sine is x, with a principal range typically between -π/2 and π/2 for arcsin. Understanding these functions is crucial for evaluating expressions involving inverse trigonometric values.
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Range of Inverse Sine Function
The range of the inverse sine function, sin⁻¹(x), is limited to the interval [-π/2, π/2]. This means that when evaluating sin⁻¹(-1), we are looking for an angle within this range whose sine value is -1. Recognizing this range helps in determining the correct angle corresponding to the given sine value.
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Unit Circle
The unit circle is a fundamental concept in trigonometry that helps visualize the values of sine and cosine for various angles. It is a circle with a radius of one centered at the origin of a coordinate plane. The sine of an angle corresponds to the y-coordinate of the point on the unit circle, which aids in understanding why sin⁻¹(-1) equals -π/2, as this is the angle where the sine value reaches its minimum.
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