91Ó°ÊÓ

Understanding the Pythagorean identity is crucial for solving many trigonometric problems. It's a fundamental linkage between the sine and cosine functions of the same angle. The identity is \( \sin^2\theta + \cos^2\theta = 1 \). In essence, it depicts that the square of the sine function and the square of the cosine function of any angle will always add up to one. This relation originates from the Pythagorean theorem applied to a unit circle. In our exercise, given that \( \cos\theta = \frac{1}{3} \) we used the Pythagorean identity to solve for \( \sin\theta \). By subtracting \( \cos^2\theta \) from both sides, we isolated \( \sin^2\theta \) and found its value to be \( \frac{8}{9} \) before taking the square root to get \( \frac{2\sqrt{2}}{3} \).

Keep in mind that \( \theta \) can be in different quadrants where sine and cosine have different signs. The quadrant of \( \theta \) helps to decide the sign of the square root when solving for \( \sin\theta \) from \( \sin^2\theta \).

The sine function is a fundamental trigonometric function that allows us to determine the vertical coordinate of an angle's unit circle representation. For an angle \( \theta \), the sine function \( \sin(\theta) \) outputs the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right-angled triangle. In the case of the unit circle, where the radius (hypotenuse) is 1, \( \sin(\theta) \) is simply the vertical coordinate of the point where the angle intersects the circle.

When we calculated \( \sin\theta \) in our exercise, we were actually finding this coordinate. With the given \( \cos\theta \) and using the Pythagorean identity, we determined \( \sin\theta \) to be \( \frac{2\sqrt{2}}{3} \) in the first quadrant, where sine is positive.

The tangent function provides a relationship between the sine and cosine functions and is particularly useful in many aspects of trigonometry. Defined as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), it represents the slope of the terminal side of the angle \( \theta \) when drawn in the coordinate system. You might associate it with the tangent of a circle, but don't confuse them; they are distinct concepts.

During the solution process for \( \tan\theta \) from the exercise, we divided the previously found \( \sin\theta \) by \( \cos\theta \), leading us to find that \( \tan\theta = 2\sqrt{2} \) without further complication. This simplification is made possible because both \( \sin\theta \) and \( \cos\theta \) have been determined already.

In contrast to \( \cos\theta \), the secant function, denoted by \( \sec\theta \), is its reciprocal. It is defined as \( \sec\theta = \frac{1}{\cos\theta} \). This function is less commonly used than sine or cosine but plays a critical role in various trigonometric problems and in the study of calculus. The secant function can sometimes provide a more direct route to a solution, particularly when we're dealing with relationships involving lengths or distances.

In our step-by-step solution, we simply took the reciprocal of \( \cos\theta \) which was given as \( \frac{1}{3} \) to find the secant. With the reciprocal identity, this calculation was straightforward, yielding \( \sec\theta = 3 \) as the result.

The cofunction identities are fascinating as they highlight symmetries in the trigonometric functions. A cofunction identity relates the trigonometric functions of complementary angles. For instance, \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta \) and \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta \) are examples of such identities. Essentially, the sine of an angle is equal to the cosine of its complement, and vice versa, because of the symmetry of right-angle triangles.

When we sought to find \( \csc(90^\circ-\theta) \) in our exercise, we used the cofunction identity to recognize that \( \csc(90^\circ-\theta) = \frac{1}{\sin(90^\circ-\theta)} = \frac{1}{\cos\theta} \) which simplifies to 3, considering our initial value of \( \cos\theta \) was \( \frac{1}{3} \) and its reciprocal is 3.