Sketch a right triangle corresponding to the trigonometric...

Sketch a right triangle corresponding to the trigonometric function of the acute angle $\theta .$ Use the Pythagorean Theorem to determine the third side of the triangle and then find the values of the other five trigonometric functions of $\theta$. $$\cos \theta=\frac{3}{7}$$

Key Concepts

Right Triangle

A right triangle is a triangle that has one 90-degree angle, and its sides are often used to illustrate and define trigonometric functions. In the context of trigonometry, one of the angles (other than the right angle) is typically designated as the angle of interest, and the sides of the triangle (opposite, adjacent, and hypotenuse) represent the relationships used to compute the trigonometric ratios.

Trigonometric Functions

Trigonometric functions (sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent) are defined based on the ratios of two sides of a right triangle. For an acute angle in 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.

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the other two sides. This fundamental theorem is used to determine the length of a missing side in a right triangle when the lengths of the other two sides are known, thereby facilitating the calculation of all trigonometric function values.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the inverses of the primary trigonometric functions. They include the secant, which is the reciprocal of the cosine; the cosecant, which is the reciprocal of the sine; and the cotangent, which is the reciprocal of the tangent. These functions are useful in various contexts, especially when dealing with equations or identities that require the use of reciprocal relationships.

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