I’m stuck bc when I draw the triangle, how do I know where theta is? The angle opposite the hypotenuse is the right Angle so I know it can’t be the angle close to the origin. I can’t start the problem w/o firsts figuring that out. Please help.
That leaves 2 other possible angles. W/o knowing the location of theta, how do I label opposite and adjacent?
Transcribed Image Text:**Trigonometric Values from a Point on the Terminal Side of an Angle**
Given the point \( (3, -\sqrt{7}) \) on the terminal side of angle \(\theta\), find the exact values of \(\cos \theta\), \(\csc \theta\), and \(\tan \theta\).
**Steps:**
1. **Calculate the hypotenuse (r)**:
- Use the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \).
- Here, \( x = 3 \) and \( y = -\sqrt{7} \).
- \( r = \sqrt{3^2 + (-\sqrt{7})^2} = \sqrt{9 + 7} = \sqrt{16} = 4 \).
2. **Find \(\cos \theta\)**:
- \(\cos \theta = \frac{x}{r} = \frac{3}{4}\).
3. **Find \(\csc \theta\)**:
- \(\csc \theta = \frac{r}{y} = \frac{4}{-\sqrt{7}} = -\frac{4}{\sqrt{7}}\).
- Rationalize the denominator: \(-\frac{4\sqrt{7}}{7}\).
4. **Find \(\tan \theta\)**:
- \(\tan \theta = \frac{y}{x} = \frac{-\sqrt{7}}{3}\).
**Conclusion:**
- \(\cos \theta = \frac{3}{4}\)
- \(\csc \theta = -\frac{4\sqrt{7}}{7}\)
- \(\tan \theta = \frac{-\sqrt{7}}{3}\)
Figure in plane geometry formed by two rays or lines that share a common endpoint, called the vertex. The angle is measured in degrees using a protractor. The different types of angles are acute, obtuse, right, straight, and reflex.
Expert Solution
Step 1
Given:
Let be a point on the terminal side of .
To find : Exact values of and .
Terminal side is defined as the side which is opposite to right angle in a right-angle triangle.
Pythagoras theorem in a right angle triangle is given as:
Where, is hypotenuse and are other sides of the right-angle triangle.
Trigonometric ratios define the relation between the sides and angles of a right angle triangle.
There are six trigonometric ratios: sine, cosine, tangent, cosecant, secant and cotangent.
