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The Pythagorean identity is a fundamental concept in trigonometry, and it relates the squares of the primary trigonometric functions — sine and cosine. The identity is expressed as:\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]This identity holds true for any angle \( \theta \). It is a direct consequence of the Pythagorean theorem applied to the unit circle where the hypotenuse has a length of one. You can think of this identity as demonstrating the inherent relationship between sine and cosine. Given one of these functions, the identity allows us to find the other.For example, if we know that \( \sin(\theta) = \frac{5}{13} \), we can find \( \cos(\theta) \) by rearranging the Pythagorean identity:\[ \cos^2(\theta) = 1 - \sin^2(\theta) \]Plugging in the value of \( \sin(\theta) \), we have:\[ \cos^2(\theta) = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \]Thus, \( \cos(\theta) = \frac{12}{13} \) or \( \cos(\theta) = -\frac{12}{13} \) depending on the quadrant. For angles within \( \frac{\pi}{2} < \theta < \pi \), \( \cos(\theta) \) is negative, so \( \cos(\theta) = -\frac{12}{13} \). The Pythagorean identity is crucial in solving trigonometric equations and transforming expressions.

Half-angle identities allow us to determine the trigonometric functions of half-angles (angles halved in value). These identities are particularly useful for breaking down complex angle problems into smaller, more manageable parts. The half-angle identities for sine, cosine, and tangent are derived from the double angle identities and are given as follows:

• \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)
• \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \)

The sign depends on the quadrant where the half-angle lies. For example, in the first quadrant, both sine and cosine half-angles are positive.
The step-by-step solution to calculate \( \sin\left(\frac{\theta}{2}\right) \) and \( \cos\left(\frac{\theta}{2}\right) \) using these half-angle identities shows us this dependence on the known value of \( \cos(\theta) \). Knowing \( \cos(\theta) = -\frac{12}{13} \), we compute:\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{25}{26}} = \frac{5}{\sqrt{26}} \]\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1}{26}} = \frac{1}{\sqrt{26}} \]These identities and calculations simplify trigonometric analysis by reducing angles, ensuring accuracy and clarity.

Trigonometric functions are the building blocks of trigonometry and include sine, cosine, and tangent, along with their reciprocals—cosecant, secant, and cotangent. These functions are defined based on the ratios of the sides of a right triangle or the coordinates of a point on the unit circle.**Sine and Cosine Functions**- The sine of an angle \(\theta\), written as \( \sin(\theta) \), is the ratio of the opposite side to the hypotenuse in a right triangle.- The cosine of \(\theta\), written as \( \cos(\theta) \), is the ratio of the adjacent side to the hypotenuse.**Tangent Function**- The tangent of \(\theta\), \( \tan(\theta) \), is the ratio of the sine and cosine functions: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).In the given exercise, these functions are used to find exact trigonometric values using identities and known values. For example, with \( \sin(\theta) = \frac{5}{13} \) and \( \cos(\theta) = -\frac{12}{13} \) in the second quadrant:

• \( \tan(\theta) = \frac{5}{-12} = -\frac{5}{12} \)
• Double angle identities such as \( -\sin(2\theta) = -2 \sin(\theta) \cos(\theta) \) then help solve for doubled angles.

Thus, mastery of trigonometric functions and their interrelations enables solving complex trigonometric problems with confidence.