Express as a function of a DIFFERENT angle, $0^{\circ} \leq \theta<360^{\circ}$ . $\newline$ $\tan \left(225^{\circ}\right)$ $\newline$ $\tan \left(\square^{\circ}\right)$

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Algebra 2

Find trigonometric ratios using reference angles

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Q. Express as a function of a DIFFERENT angle,

0^{\circ} \leq \theta<360^{\circ}

\newline

\tan \left(225^{\circ}\right)

\newline

\tan \left(\square^{\circ}\right)

Understand Problem and Range: Understand the problem and the range for $\theta$ . We need to express $\tan(225°)$ in terms of another angle that is within the range of $0°$ to $360°$ . Since $225°$ is outside this range, we need to find an equivalent angle within the range.
Find Reference Angle: Find the reference angle for $225°$ . $\newline$ The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. Since $225°$ is in the third quadrant, its reference angle is $225° - 180° = 45°$ .
Determine Tangent Sign: Determine the sign of the tangent function in the third quadrant. $\newline$ In the third quadrant, both sine and cosine are negative, so the tangent (which is sine divided by cosine) is positive. Therefore, $\tan(225°)$ will have the same value as $\tan(45°)$ but with a positive sign.
Express as Function of Reference Angle: Express $\tan(225°)$ as a function of the reference angle. $\newline$ Since the reference angle is $45°$ and $\tan(45°) = 1$ , we can express $\tan(225°)$ as $\tan(45°)$ . However, we need to find a different angle, not the reference angle itself.
Find Angle with Same Tangent Value: Find a different angle with the same tangent value. $\newline$ We know that the tangent function has a period of $180°$ , so if we add or subtract multiples of $180°$ to $45°$ , we will get angles with the same tangent value. The closest angle within the range of $0°$ to $360°$ that is not $45°$ is $45° + 180° = 225°$ , but this is the original angle given. The next angle would be $45° - 180°$ , but this is negative and outside the range. Therefore, we need to look for another approach.
Use Tangent Symmetry: Use the symmetry of the tangent function. $\newline$ The tangent function is also symmetrical about the origin, which means that $\tan(\theta) = -\tan(180° + \theta)$ . So, we can find an angle $\theta$ such that $180° + \theta = 225°$ . Solving for $\theta$ gives us $\theta = 225° - 180° = 45°$ , but again, this is the reference angle itself.
Correct Approach: Realize the mistake and correct the approach. $\newline$ We made a mistake in the previous steps by not finding a different angle. We need to find an angle that is coterminal with $225°$ and within the range of $0°$ to $360°$ . Since $225°$ is already within the range, we need to find an angle that has the same tangent value but is not the same as $225°$ or its reference angle of $45°$ .