5.2 Trigonometric Functions of Real Numbers

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5.2.1 Learning Objectives

By the end of this section, you should be able to:

• define the six trigonometric functions using terminal points;
• evaluate exact trigonometric values;
• determine signs, domains, and periods;
• use fundamental identities; and
• calculate trigonometric values from one known value.

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5.2.2 Definitions from the Unit Circle

If \(P(t)=(x,y)\), then

\[ \sin t=y,\qquad \cos t=x,\qquad \tan t=\frac yx, \]

and

\[ \csc t=\frac1y,\qquad \sec t=\frac1x,\qquad \cot t=\frac xy, \]

where the denominators are nonzero.

Example 5.5 Evaluating from a Terminal Point

If \(P(t)=(-3/5,4/5)\), find all six trigonometric values.

Solution

\[ \boxed{\sin t=\frac45,\ \cos t=-\frac35,\ \tan t=-\frac43,\ \csc t=\frac54,\ \sec t=-\frac53,\ \cot t=-\frac34} \]

Exercise 5.5 If \(P(t)=(5/13,-12/13)\), find \(\sin t\), \(\cos t\), and \(\tan t\).

Show answer

Answer: \(\boxed{\sin t=-12/13,\quad \cos t=5/13,\quad \tan t=-12/5}\)

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5.2.3 Exact Values and Signs

Exact values come directly from unit-circle coordinates. In Quadrants I through IV, the positive functions are respectively: all functions; sine and cosecant; tangent and cotangent; cosine and secant.

Example 5.6 Finding an Exact Value

Evaluate \(\sin(5\pi/3)\), \(\cos(5\pi/3)\), and \(\tan(5\pi/3)\).

Solution

The terminal point is \((1/2,-\sqrt3/2)\). Therefore,

\[ \boxed{\sin(5\pi/3)=-\frac{\sqrt3}{2},\quad \cos(5\pi/3)=\frac12,\quad \tan(5\pi/3)=-\sqrt3} \]

Exercise 5.6 Evaluate \(\sec(3\pi/4)\).

Show answer

Answer: Since \(\cos(3\pi/4)=-\sqrt2/2\), \(\boxed{\sec(3\pi/4)=-\sqrt2}\).

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5.2.4 Identities and Periodicity

The reciprocal and quotient identities are

\[ \csc t=\frac1{\sin t},\quad \sec t=\frac1{\cos t},\quad \cot t=\frac1{\tan t}, \]

\[ \tan t=\frac{\sin t}{\cos t},\qquad \cot t=\frac{\cos t}{\sin t}. \]

The Pythagorean identity is

\[ \sin^2t+\cos^2t=1. \]

Sine and cosine have period \(2\pi\); tangent and cotangent have period \(\pi\).

Example 5.7 Using the Pythagorean Identity

If \(\cos t=-5/13\) and \(t\) is in Quadrant III, find \(\sin t\).

Solution

\[ \sin^2t=1-\frac{25}{169}=\frac{144}{169}. \]

Sine is negative in Quadrant III, so

\[ \boxed{\sin t=-\frac{12}{13}}. \]

Exercise 5.7 If \(\sin t=3/5\) and \(t\) is in Quadrant II, find \(\cos t\).

Show answer

Answer: \(\boxed{\cos t=-4/5}\)

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5.2.5 Even and Odd Trigonometric Functions

Cosine and secant are even:

\[ \cos(-t)=\cos t,\qquad \sec(-t)=\sec t. \]

Sine, cosecant, tangent, and cotangent are odd, so \(f(-t)=-f(t)\).

Example 5.8 Using Symmetry

Evaluate \(\sin(-\pi/6)\) and \(\cos(-\pi/3)\).

Solution

\[ \boxed{\sin(-\pi/6)=-1/2,\qquad \cos(-\pi/3)=1/2} \]

Exercise 5.8 Evaluate \(\tan(-\pi/4)\).

Show answer

Answer: \(\boxed{-1}\)

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5.2.6 Conceptual Takeaways

• Unit-circle coordinates define sine and cosine.
• The other four functions are ratios or reciprocals.
• Quadrants determine signs.
• Identities connect the six functions.
• Periodicity and symmetry reduce unfamiliar inputs to familiar ones.

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5.2.7 Skills You Should Be Able to Do

• Evaluate all six trigonometric functions.
• Determine exact values from reference numbers.
• Identify undefined values.
• Apply identities and periodicity.
• Use one value and a quadrant to find the others.

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5.2.8 Practice Problems with Solutions

1. Evaluate \(\sin(\pi/4)\) and \(\cos(\pi/4)\).

   Show Solution

   The terminal point is \((\sqrt2/2,\sqrt2/2)\), so

   \[ \boxed{\sin(\pi/4)=\cos(\pi/4)=\sqrt2/2}. \]

2. Evaluate \(\tan(2\pi/3)\).

   Show Solution

   \[ \tan(2\pi/3)=\frac{\sqrt3/2}{-1/2}=\boxed{-\sqrt3}. \]

3. State where \(\sec t\) is undefined.

   Show Solution

   Secant is undefined when \(\cos t=0\):

   \[ \boxed{t=\pi/2+k\pi,\quad k\in\mathbb Z}. \]

4. If \(\sin t=-8/17\) and \(t\) is in Quadrant IV, find \(\cos t\).

   Show Solution

   \[ \cos^2t=1-\frac{64}{289}=\frac{225}{289}. \]

   Cosine is positive in Quadrant IV, so \(\boxed{\cos t=15/17}\).

5. Evaluate \(\cos(13\pi/6)\).

   Show Solution

   Since \(13\pi/6=2\pi+\pi/6\),

   \[ \boxed{\cos(13\pi/6)=\sqrt3/2}. \]

6. If \(\tan t=2\) and \(t\) is in Quadrant III, determine the signs of all six functions.

   Show Solution

   In Quadrant III, sine and cosine are negative, while tangent is positive. Reciprocal functions have the same signs as their partners.

   \[ \boxed{\sin,\cos,\csc,\sec<0;\qquad \tan,\cot>0} \]

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