Sine and Cosine Functions: Fresh Take

  • Find function values for the sine and cosine of the special angles.
  • Use reference angles to evaluate trigonometric functions.
  • Evaluate sine and cosine values using a calculator.

Finding Function Values

The Main Idea

Some angles come up so often in trigonometry that we call them the special angles: [latex]30^\circ[/latex], [latex]45^\circ[/latex], and [latex]60^\circ[/latex] (or [latex]\dfrac{\pi}{6}[/latex], [latex]\dfrac{\pi}{4}[/latex], and [latex]\dfrac{\pi}{3}[/latex] in radians). The sine and cosine values of these angles can be found using right triangles or the unit circle. Instead of memorizing long lists, you can use patterns and symmetry to recall them quickly. These special values form the backbone of the unit circle and are used throughout trigonometry.

Quick Tips: Special Sine and Cosine Values

  1. The 30–60–90 Triangle:

    • [latex]\sin(30^\circ)=\dfrac{1}{2}[/latex], [latex]\cos(30^\circ)=\dfrac{\sqrt{3}}{2}[/latex]

    • [latex]\sin(60^\circ)=\dfrac{\sqrt{3}}{2}[/latex], [latex]\cos(60^\circ)=\dfrac{1}{2}[/latex]

  2. The 45–45–90 Triangle:

    • [latex]\sin(45^\circ)=\cos(45^\circ)=\dfrac{\sqrt{2}}{2}[/latex]

  3. Unit Circle Reminder: On the unit circle, sine = y-coordinate, cosine = x-coordinate.

  4. Pattern Trick: For sine of 30°, 45°, 60°, think [latex]\dfrac{1}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}[/latex]. For cosine, the order reverses.

  5. Radians Are Your Friend: Know the same values in radian form: [latex]\dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}[/latex].

 

A “perfect pizza slice” trend suggests cutting the pie into [latex]6[/latex] equal slices. The center line to a slice edge makes an angle of [latex]60^\circ[/latex]. Find [latex]\sin(60^\circ)[/latex] and [latex]\cos(60^\circ)[/latex].

 

A viral “speed-quilt” block uses a diagonal cut at [latex]45^\circ[/latex]. Evaluate [latex]\sin(45^\circ)[/latex] and [latex]\cos(45^\circ)[/latex].

 

At a community festival drum circle, Section A sits at [latex]0^\circ[/latex] and Section C sits at [latex]120^\circ[/latex]. Find [latex]\sin(120^\circ)[/latex] and [latex]\cos(120^\circ)[/latex].

 

Using Reference Angles

The Main Idea

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. Reference angles are useful because they let us connect any angle on the unit circle back to one of the familiar “special angles” ([latex]30^\circ[/latex], [latex]45^\circ[/latex], [latex]60^\circ[/latex]). By knowing the sine and cosine values for these special angles and then adjusting the sign based on the quadrant, we can evaluate trig functions for almost any angle without a calculator.

Quick Tips: 

  1. Find the Reference Angle: First, find which quadrant the angle is in and then use the reference angle rule that applies:
    Quadrant Degrees Radians

    I

    		 given angle given angle
    II [latex]180^\circ -[/latex] given angle [latex]\pi -[/latex] given angle
    III given angle [latex]- 180^\circ[/latex] given angle [latex]- \pi[/latex]
    IV [latex]360^\circ -[/latex] given angle [latex]2\pi -[/latex] given angle
  2. Use Special Angle Values: Match the reference angle to [latex]30^\circ[/latex], [latex]45^\circ[/latex], or [latex]60^\circ[/latex] (or their radian equivalents) to recall sine and cosine values.
  3. Apply Quadrant Signs:
    Quadrant [latex]\cos[/latex] [latex]\sin[/latex]

    I

    		 [latex]+[/latex] [latex]+[/latex]
    II [latex]-[/latex] [latex]+[/latex]
    III [latex]-[/latex] [latex]-[/latex]
    IV [latex]+[/latex] [latex]-[/latex]

  4. General Strategy: “Reference angle gives the value, quadrant gives the sign.”

  5. Check with Examples:

    • Example 1: [latex]\sin(150^\circ)[/latex]

      • Step 1: Find the reference angle → [latex]180^\circ - 150^\circ = 30^\circ[/latex].

      • Step 2: Recall [latex]\sin(30^\circ) = \dfrac{1}{2}[/latex].

      • Step 3: Determine the quadrant → [latex]150^\circ[/latex] is in Quadrant II, where sine values are positive.

      • Answer: [latex]\sin(150^\circ) = \dfrac{1}{2}[/latex].

    • Example 2: [latex]\cos(\dfrac{7\pi}{6})[/latex]

      • Step 1: Find the reference angle → [latex]\dfrac{7\pi}{6} - \pi = \dfrac{\pi}{6}[/latex].

      • Step 2: Recall [latex]\cos (\dfrac{\pi}{6}) = \dfrac{\sqrt{3}}{2}[/latex].

      • Step 3: Determine the quadrant → [latex]\dfrac{7\pi}{6}[/latex] is in Quadrant III, where cosine values are negative.

      • Answer: [latex]\cos(\dfrac{7\pi}{6}) = - \dfrac{\sqrt{3}}{2}[/latex].

 

A transit map shows a detour bearing of [latex]-210^\circ[/latex]. Use a reference angle to find [latex]\sin(-210^\circ)[/latex] and [latex]\cos(-210^\circ).[/latex]

 

A viral line dance calls for a [latex]765^\circ[/latex] spin. Use a reference angle to evaluate [latex]\sin(765^\circ)[/latex] and [latex]\cos(765^\circ)[/latex].

 

Turning a tight lid, you rotate the handle by [latex]\dfrac{23\pi}{6}[/latex]. Using a reference angle, find [latex]\sin\dfrac{23\pi}{6}[/latex] and [latex]\cos\dfrac{23\pi}{6}[/latex].

 

Evaluating Sine and Cosine Values

The Main Idea

While special angles can be evaluated exactly, most angles require a calculator. A calculator lets us find approximate values of sine and cosine for any angle, but it’s important to use the correct mode (degrees or radians) depending on how the angle is given. Understanding how to set up your calculator and interpret its output ensures accurate results and prevents common mistakes.

Quick Tips:  Using a Calculator for Sine and Cosine

  1. Check the Mode First:

    • If the angle is in degrees, set the calculator to degree mode.

    • If the angle is in radians, set the calculator to radian mode.

  2. Enter the Angle Directly: Use [latex]\sin(\theta)[/latex] or [latex]\cos(\theta)[/latex] and press enter.

  3. Expect Decimals: Calculators return approximate values (e.g., [latex]\sin(40^\circ) \approx 0.6428[/latex]).

  4. Round Smartly: Round answers to 3–4 decimal places unless more precision is needed.

  5. Estimate First: Compare to a nearby special angle so the result makes sense (e.g., [latex]\sin(40^\circ)[/latex] should be close to [latex]\sin(45^\circ)=\dfrac{\sqrt{2}}{2}\approx0.7071[/latex]).

  6. Common Pitfall: Wrong mode = wrong answer. If your sine or cosine seems “way off,” double-check the calculator’s mode.

 

A stadium “wave” model uses [latex]t = 1.12[/latex] radians as the phase. Compute [latex]\sin(1.12)[/latex] and [latex]\cos(1.12)[/latex] and round to [latex]4[/latex] decimals.

 

Your trail app shows a bearing of [latex]247^\circ[/latex]. Find [latex]\sin(247^\circ)[/latex] and [latex]\cos(247^\circ)[/latex] and round to [latex]4[/latex] decimals.

 

A video game controller’s thumbstick reads [latex]2.87[/latex] radians for a circle-dash move. Compute [latex]\sin(2.87)[/latex] and [latex]\cos(2.87)[/latex] and round to [latex]4[/latex] decimals.