Concept Check Suppose that 90° < θ < 180° .   Find the sign of each function value. tan θ/2

Verified step by step guidance

1. <insert step 1> Identify the range of \( \theta \) given as \( 90^\circ < \theta < 180^\circ \). This places \( \theta \) in the second quadrant of the unit circle.>
2. <insert step 2> Recall that in the second quadrant, the sine function is positive, and the cosine function is negative.>
3. <insert step 3> Use the identity for the tangent of half an angle: \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \).>
4. <insert step 4> Since \( \cos \theta \) is negative in the second quadrant, the expression \( 1 - \cos \theta \) is positive, and \( 1 + \cos \theta \) is less than 1 but positive.>
5. <insert step 5> Therefore, \( \tan \frac{\theta}{2} \) is positive because the square root of a positive number is positive.>

Key Concepts

Quadrants of Trigonometric Functions

In trigonometry, the unit circle is divided into four quadrants, each corresponding to specific ranges of angles. For angles between 90° and 180° (the second quadrant), sine is positive while cosine and tangent are negative. Understanding which functions are positive or negative in each quadrant is essential for determining the sign of trigonometric function values.

Half-Angle Formulas

The half-angle formulas are used to find the sine, cosine, and tangent of half of a given angle. For tangent, the formula is tan(θ/2) = sin(θ)/(1 + cos(θ)). This is particularly useful when analyzing angles that are not standard, as it allows us to relate the function values of the original angle to those of its half.

Sign of Tangent in the Second Quadrant

In the second quadrant, where 90° < θ < 180°, the tangent function is negative because it is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Since sine is positive and cosine is negative in this quadrant, the overall value of tangent becomes negative. This understanding is crucial for determining the sign of tan(θ/2) when θ is in the specified range.