Given that θ=tan^-1((4)/(3)), find the exact values of sinθ cosθ, cotθ, secθ, and

step 1 Okay, we are giving that an angle equals the inverse tangent of four thirds. So what that really

Given that θ=tan^-1((4)/(3)), find the exact values...

$$\begin{aligned}
&\text { Given that } \theta=\tan ^{-1}\left(\frac{4}{3}\right), \text { find the exact values of } \sin \theta\\
&\cos \theta, \cot \theta, \sec \theta, \text { and } \csc \theta
\end{aligned}$$

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine an angle when a trigonometric ratio is known. For instance, using tan^(-1) allows one to find an angle given its tangent value, establishing a connection between the ratio and the actual angle measure.

Right Triangle Constructions and the Pythagorean Theorem

When given the value of a trigonometric ratio, one can construct a right triangle where the sides represent the components of that ratio. The Pythagorean theorem is then employed to determine the missing side, providing a framework to compute all six trigonometric functions.

Trigonometric Function Definitions

Trigonometric functions such as sine, cosine, and tangent are defined in the context of a right triangle. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. These definitions allow translation between an angle’s value and the ratios of the sides of a right triangle.

Reciprocal and Quotient Identities

Reciprocal identities (like secant being the reciprocal of cosine, and cosecant the reciprocal of sine) and quotient identities (such as cotangent being the reciprocal of tangent) are fundamental in trigonometry. They provide relationships among the trigonometric functions, enabling the calculation of one function from another and deepening understanding of their interconnections.

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