Textbook Question
Determine whether each statement is possible or impossible. a. sec θ = ―2/3
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
3:51 minutesProblem 46
Textbook Question
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.
csc θ > 0 , cot θ > 0
Verified step by step guidance1
Recall the definitions and signs of the trigonometric functions in each quadrant. The cosecant function, \(\csc \theta\), is the reciprocal of sine, so \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc \theta > 0\) means \(\sin \theta > 0\).
Determine in which quadrants \(\sin \theta\) is positive. Since sine corresponds to the y-coordinate on the unit circle, \(\sin \theta > 0\) in Quadrant I and Quadrant II.
Next, analyze the condition \(\cot \theta > 0\). Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). For \(\cot \theta\) to be positive, the ratio \(\frac{\cos \theta}{\sin \theta}\) must be positive, meaning \(\cos \theta\) and \(\sin \theta\) have the same sign.
Since from step 2, \(\sin \theta > 0\), for \(\cot \theta\) to be positive, \(\cos \theta\) must also be positive. Identify the quadrants where both \(\sin \theta\) and \(\cos \theta\) are positive.
Conclude that the angle \(\theta\) must lie in Quadrant I, where both sine and cosine are positive, satisfying both \(\csc \theta > 0\) and \(\cot \theta > 0\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Signs of Trigonometric Functions in Quadrants
The signs of sine, cosine, tangent, and their reciprocals vary depending on the quadrant of the angle. For example, sine and cosecant are positive in Quadrants I and II, while cotangent is positive in Quadrants I and III. Understanding these sign patterns helps determine the possible quadrants for an angle.
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Reciprocal Trigonometric Functions
Cosecant (csc) is the reciprocal of sine (sin), and cotangent (cot) is the reciprocal of tangent (tan). Knowing these relationships allows you to infer the sign of sine and tangent from the signs of cosecant and cotangent, which is essential for identifying the correct quadrant.
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Quadrant Determination Using Multiple Conditions
When given multiple inequalities involving trigonometric functions, the solution involves finding the intersection of quadrants where all conditions hold true. This requires combining the sign information of each function to narrow down the possible quadrants for the angle θ.
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