6.3 Trigonometric Functions of Angles
6.3.1 Learning Objectives
By the end of this section, you should be able to:
- place angles in standard position;
- find reference angles;
- evaluate trigonometric functions for general angles;
- determine signs by quadrant; and
- use coterminal angles and identities.
6.3.2 Angles in Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies on the positive \(x\)-axis. Positive angles rotate counterclockwise and negative angles rotate clockwise.
Example 6.8 Locating a Quadrant
In which quadrant does \(230^\circ\) terminate?
Solution
Since \(180^\circ<230^\circ<270^\circ\),
\[ \boxed{\text{Quadrant III}}. \]
Exercise 6.8 In which quadrant does \(-50^\circ\) terminate?
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Answer: \(\boxed{\text{Quadrant IV}}\)
6.3.3 Definitions for General Angles
If \((x,y)\) lies on the terminal side and \(r=\sqrt{x^2+y^2}\), then
\[ \sin\theta=\frac yr,\quad \cos\theta=\frac xr,\quad \tan\theta=\frac yx, \]
with reciprocal definitions for cosecant, secant, and cotangent.
Example 6.9 Evaluating from a Terminal-Side Point
The terminal side contains \((-8,15)\). Find \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
Solution
\[ r=\sqrt{(-8)^2+15^2}=17. \]
\[ \boxed{\sin\theta=15/17,\quad \cos\theta=-8/17,\quad \tan\theta=-15/8} \]
Exercise 6.9 The terminal side contains \((5,-12)\). Find \(\cos\theta\).
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Answer: Since \(r=13\), \(\boxed{\cos\theta=5/13}\).
6.3.4 Reference Angles and Exact Values
The reference angle is the acute angle between the terminal side and the \(x\)-axis. Use it for the magnitude of a trigonometric value, then apply the quadrant sign.
Example 6.10 Using a Reference Angle
Evaluate \(\sin210^\circ\) and \(\cos210^\circ\).
Solution
The reference angle is \(30^\circ\). Both sine and cosine are negative in Quadrant III:
\[ \boxed{\sin210^\circ=-1/2,\quad \cos210^\circ=-\sqrt3/2}. \]
Exercise 6.10 Evaluate \(\tan135^\circ\).
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Answer: The reference angle is \(45^\circ\), and tangent is negative in Quadrant II, so \(\boxed{-1}\).
6.3.5 Conceptual Takeaways
- Standard position gives a consistent geometric meaning to any angle.
- The radius \(r\) is always positive.
- Reference angles determine magnitudes.
- Quadrants determine signs.
- Coterminal angles have identical trigonometric values.
6.3.6 Skills You Should Be Able to Do
- Sketch angles in standard position.
- Identify quadrants and reference angles.
- Evaluate functions from terminal-side points.
- Find exact values for common angles.
- Apply signs and coterminal relationships.
6.3.7 Practice Problems with Solutions
Find the quadrant and reference angle for \(320^\circ\).
Show Solution
The angle is in Quadrant IV, and \(360^\circ-320^\circ=40^\circ\).
\[ \boxed{\text{Quadrant IV; reference angle }40^\circ} \]
Evaluate \(\sin150^\circ\).
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The reference angle is \(30^\circ\), and sine is positive in Quadrant II:
\[ \boxed{1/2}. \]
Evaluate \(\cos240^\circ\).
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The reference angle is \(60^\circ\), and cosine is negative in Quadrant III:
\[ \boxed{-1/2}. \]
Evaluate \(\tan315^\circ\).
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The reference angle is \(45^\circ\), and tangent is negative in Quadrant IV:
\[ \boxed{-1}. \]
A terminal side contains \((-7,-24)\). Find all six trigonometric values.
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Since \(r=25\),
\[ \boxed{\sin=-24/25,\ \cos=-7/25,\ \tan=24/7,\ \csc=-25/24,\ \sec=-25/7,\ \cot=7/24}. \]
Use Desmos to verify that \(\sin(-110^\circ)=\sin250^\circ\). Explain why.
Show Solution
The angles differ by \(360^\circ\), so they are coterminal and have the same terminal side.
\[ \boxed{\sin(-110^\circ)=\sin250^\circ}. \]