Reference Angles: Videos & Practice Problems

Understanding reference angles is crucial for determining trigonometric values across all quadrants of the unit circle. Each angle corresponds to a reference angle in quadrant 1, allowing for quick calculations. The sine, cosine, and tangent values of these angles mirror those of their reference angles, with sign changes based on quadrant location. For example, in quadrant 2, only sine is positive, while in quadrant 3, tangent is positive. Recognizing coterminal angles, such as 390 degrees equating to 30 degrees, further simplifies finding trigonometric values.

Reference Angles on the Unit Circle

Understanding trigonometric values for angles beyond the first quadrant can initially seem daunting, but the concept of reference angles simplifies this process significantly. A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis, which allows us to relate angles in other quadrants back to those in the first quadrant, where we already know the trigonometric values.

To find the reference angle for any angle not in the first quadrant, measure the angle to the nearest x-axis. For example, for an angle of 150 degrees, the nearest x-axis is at 180 degrees. The difference between these two angles is 30 degrees, making the reference angle for 150 degrees equal to 30 degrees. Similarly, for 135 degrees, the nearest x-axis is again 180 degrees, and the difference is 45 degrees, giving it a reference angle of 45 degrees. For 120 degrees, the nearest x-axis is still 180 degrees, and the difference is 60 degrees, resulting in a reference angle of 60 degrees.

In the third quadrant, for an angle of 210 degrees, the nearest x-axis is 180 degrees, and the difference is again 30 degrees, so the reference angle is 30 degrees. For 330 degrees in the fourth quadrant, the nearest x-axis is 360 degrees, and the difference is also 30 degrees, making its reference angle 30 degrees as well. For angles like 225 degrees and 315 degrees, both are 45 degrees away from their nearest x-axis, giving them a reference angle of 45 degrees. Lastly, for 240 degrees and 300 degrees, both are 60 degrees away from their respective nearest x-axes, resulting in a reference angle of 60 degrees.

To help remember these reference angles, one can visualize them forming an "X" shape. All angles with a reference angle of 30 degrees create one "X," while those with a reference angle of 45 degrees form another, and angles with a reference angle of 60 degrees create a taller, thinner "X." Additionally, when considering radian measures, angles with a reference angle of 30 degrees share a denominator of 6, those with 45 degrees share a denominator of 4, and those with 60 degrees share a denominator of 3.

By mastering the concept of reference angles, students can efficiently determine trigonometric values for a wide range of angles across all four quadrants, reinforcing their understanding of the unit circle and the relationships between angles.

Identify the reference angle of each given angle.
120°

• A. 30°
• B. 45°
• C. 60°

Answer: (C)

Identify the reference angle of each given angle.
$\frac{7\pi}{4}$ rad

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$

Answer: (B)

Identify the reference angle of each given angle.
210°

• A. 30°
• B. 45°
• C. 60°

Answer: (A)

Trig Values in Quadrants II, III, & IV

Understanding the trigonometric values of angles in different quadrants is essential for mastering trigonometry. The key takeaway is that the sine, cosine, and tangent values of angles in quadrants 2, 3, and 4 can be derived from the reference angles in quadrant 1, with adjustments made for their signs based on the quadrant's characteristics.

In quadrant 1, all trigonometric values are positive. As we move to quadrant 2, for example, the angle of 150 degrees has a reference angle of 30 degrees. The cosine and sine values remain the same as those of the reference angle, but the cosine becomes negative because it corresponds to the x-coordinate, which is negative in this quadrant. Thus, for 150 degrees, the values are:

Cosine: \( -\frac{\sqrt{3}}{2} \), Sine: \( \frac{1}{2} \), Tangent: \( -\frac{\sqrt{3}}{3} \).

In quadrant 3, consider the angle of 225 degrees, which has a reference angle of 45 degrees. Here, both the sine and cosine values are negative, while the tangent remains positive since it is the ratio of sine to cosine:

Cosine: \( -\frac{\sqrt{2}}{2} \), Sine: \( -\frac{\sqrt{2}}{2} \), Tangent: \( 1 \).

Moving to quadrant 4, for the angle of 300 degrees with a reference angle of 60 degrees, the cosine is positive while the sine is negative, leading to:

Cosine: \( \frac{1}{2} \), Sine: \( -\frac{\sqrt{3}}{2} \), Tangent: \( -\frac{\sqrt{3}}{3} \).

To remember which trigonometric functions are positive in each quadrant, a helpful mnemonic is "All Students Take Calculus." This indicates that in quadrant 1, all functions are positive; in quadrant 2, only sine is positive; in quadrant 3, only tangent is positive; and in quadrant 4, only cosine is positive.

By understanding these relationships and the adjustments needed for each quadrant, you can effectively navigate the unit circle and determine the trigonometric values for any angle.

Example 1

Understanding reference angles is crucial for determining the values of trigonometric functions in different quadrants of the unit circle. The key concept is that the trigonometric values in quadrants 2, 3, and 4 can be derived from their corresponding values in quadrant 1, with adjustments made for the signs based on the quadrant's characteristics.

To remember which trigonometric functions are positive in each quadrant, the mnemonic "All Students Take Calculus" is helpful. This indicates that:

• Quadrant I: All functions are positive.
• Quadrant II: Sine is positive.
• Quadrant III: Tangent is positive.
• Quadrant IV: Cosine is positive.

In quadrant 2, for angles 135 degrees and 120 degrees, the reference angles are 45 degrees and 60 degrees, respectively. The corresponding trigonometric values from quadrant 1 are:

• For 135 degrees:
  • Cosine: \(-\frac{\sqrt{2}}{2}\)
  • Sine: \(\frac{\sqrt{2}}{2}\)
  • Tangent: \(-1\)
• For 120 degrees:
  • Cosine: \(-\frac{1}{2}\)
  • Sine: \(\frac{\sqrt{3}}{2}\)
  • Tangent: \(-\frac{\sqrt{3}}{3}\)

Moving to quadrant 3, for angles 210 degrees and 240 degrees, the reference angles are 30 degrees and 60 degrees. The values are:

• For 210 degrees:
  • Cosine: \(-\frac{\sqrt{3}}{2}\)
  • Sine: \(-\frac{1}{2}\)
  • Tangent: \(\frac{\sqrt{3}}{3}\)
• For 240 degrees:
  • Cosine: \(-\frac{1}{2}\)
  • Sine: \(-\frac{\sqrt{3}}{2}\)
  • Tangent: \(\sqrt{3}\)

In quadrant 4, for angles 330 degrees and 315 degrees, the reference angles are 30 degrees and 45 degrees. The values are:

• For 330 degrees:
  • Cosine: \(\frac{\sqrt{3}}{2}\)
  • Sine: \(-\frac{1}{2}\)
  • Tangent: \(-\frac{\sqrt{3}}{3}\)
• For 315 degrees:
  • Cosine: \(\frac{\sqrt{2}}{2}\)
  • Sine: \(-\frac{\sqrt{2}}{2}\)
  • Tangent: -1

By applying these principles, you can effectively determine the trigonometric values for any angle in the unit circle by using reference angles and adjusting the signs according to the quadrant. Practice these concepts to reinforce your understanding and improve your skills in trigonometry.

Identify what angle, $\theta$ , satisfies the following conditions.
$\sin\theta=\frac12$; $\tan\theta$ < $0$

• A. 30°
• B. 150°
• C. 60°
• D. 300°

Answer: (B)

Identify what angle, $\theta$ , satisfies the following conditions.
$\cos\theta=\frac{\sqrt3}{2}$; $\sin\theta$ < $0$

• A. 30°
• B. 60°
• C. 120°
• D. 330°

Answer: (D)

Identify what angle, $\theta$ , satisfies the following conditions.
$\tan\theta=-1$; $\cos\theta$ > $0$

• A. 45°
• B. 135°
• C. 315°
• D. 330°

Answer: (C)

Example 2

Understanding the unit circle is essential for mastering trigonometry, as it provides a visual representation of the relationships between angles and their corresponding sine, cosine, and tangent values. The unit circle is centered at the origin (0,0) with a radius of 1, and it is divided into four quadrants, each containing specific angle measures in both degrees and radians.

To begin filling in the unit circle, start with the four quadrantal angles: 0 degrees (0 radians), 90 degrees (π/2 radians), 180 degrees (π radians), and 270 degrees (3π/2 radians). Completing these angles provides a foundation for understanding the circle's structure. The coordinates for these angles are as follows: (1, 0) for 0 degrees, (0, 1) for 90 degrees, (-1, 0) for 180 degrees, and (0, -1) for 270 degrees. The tangent values, calculated as y/x, yield 0 for 0 and 180 degrees, and undefined for 90 and 270 degrees.

Next, focus on the first quadrant, where the common angles are 30 degrees (π/6), 45 degrees (π/4), and 60 degrees (π/3). The reference angle concept is crucial here, as it allows you to determine the corresponding angles in the other quadrants. For example, in the second quadrant, the angle 150 degrees (5π/6) is derived from subtracting the reference angle from 180 degrees. In the third quadrant, the angle 210 degrees (7π/6) is found by adding the reference angle to 180 degrees. Finally, in the fourth quadrant, the angle 330 degrees (11π/6) is calculated by subtracting the reference angle from 360 degrees.

When filling in the trigonometric values, remember that in the first quadrant, all values are positive. The sine, cosine, and tangent values for the angles can be derived from the coordinates of the points on the unit circle. For instance, for 30 degrees, the sine is 1/2, cosine is √3/2, and tangent is √3/3. As you move to the other quadrants, the signs of these values change based on the quadrant's characteristics. The mnemonic "All Students Take Calculus" helps remember which trigonometric functions are positive in each quadrant: sine is positive in the second quadrant, tangent in the third, and cosine in the fourth.

By systematically filling in the unit circle and applying the reference angle method, along with understanding the signs of the trigonometric functions in each quadrant, you can effectively master the unit circle. Repetition and practice will enhance your familiarity with these concepts, making it easier to recall the values and their relationships in future mathematical applications.

Coterminal Angles on the Unit Circle

Understanding coterminal angles is essential for evaluating trigonometric functions efficiently. A coterminal angle is defined as an angle that shares the same terminal side as another angle, which can be found by adding or subtracting multiples of \(360^\circ\) or \(2\pi\) radians. For instance, to find the sine of \(390^\circ\), you can subtract \(360^\circ\) to identify its coterminal angle, which is \(30^\circ\). Since both angles are coterminal, their sine values are identical. Thus, the sine of \(390^\circ\) is the same as the sine of \(30^\circ\), which is \(\frac{1}{2}\).

Similarly, when evaluating the tangent of \(3\pi\), you can visualize this on the unit circle. A full rotation of \(2\pi\) radians plus an additional \(\pi\) radians brings you back to the angle \(\pi\). Therefore, the tangent of \(3\pi\) is equivalent to the tangent of \(\pi\), which is \(0\).

For negative angles, such as \(-\frac{\pi}{4}\), the direction of measurement is clockwise. By moving clockwise from \(0\) to \(-\frac{\pi}{4}\), you arrive at the coterminal angle of \(\frac{7\pi}{4}\). The cosine of \(-\frac{\pi}{4}\) can then be determined by finding the cosine of \(\frac{7\pi}{4}\), which is \(\frac{\sqrt{2}}{2}\).

In summary, recognizing coterminal angles allows for quick identification of trigonometric values on the unit circle, simplifying the process of evaluating functions for both positive and negative angles. This understanding is crucial for mastering trigonometry and its applications.

For each expression, identify which coterminal angle to use & determine the exact value of the expression.
$\sin\frac{7\pi}{3}$

• A. $\frac12$
• B. $2$
• C. $\frac{\sqrt3}{2}$
• D. $\frac{2\sqrt3}{3}$

Answer: (C)

For each expression, identify which coterminal angle to use & determine the exact value of the expression.
$\tan765\degree$

• A. $-1$
• B. $1$
• C. $0$
• D. undefined

Answer: (B)

For each expression, identify which coterminal angle to use & determine the exact value of the expression.
$\cos\left(-\frac{10\pi}{4}\right)$

• A. $0$
• B. $1$
• C. $-1$
• D. $\frac{\sqrt2}{2}$

Answer: (A)

Here’s what students ask on this topic:

What is a reference angle and how do you find it?

A reference angle is the smallest angle between the terminal side of a given angle and the x-axis. It is always a positive acute angle (less than 90 degrees). To find the reference angle, you measure the angle to the nearest x-axis. For example, if you have an angle of 150 degrees, you subtract it from 180 degrees (the nearest x-axis), giving you a reference angle of 30 degrees. In radians, if you have an angle of 5π/6, you subtract it from π, resulting in a reference angle of π/6.

How do reference angles help in finding trigonometric values?

Reference angles simplify finding trigonometric values for angles in any quadrant. The trigonometric values (sine, cosine, tangent) of an angle are the same as those of its reference angle, with the only difference being the sign. The sign depends on the quadrant in which the original angle is located. For instance, in quadrant 2, sine is positive while cosine and tangent are negative. This allows you to use known values from quadrant 1 to quickly determine the values in other quadrants.

What are coterminal angles and how do you find them?

Coterminal angles are angles that share the same terminal side but differ by full rotations of 360 degrees (or 2π radians). To find coterminal angles, you add or subtract multiples of 360 degrees (or 2π radians). For example, to find a coterminal angle of 390 degrees, you subtract 360 degrees, resulting in 30 degrees. Similarly, for an angle of -45 degrees, you add 360 degrees to get 315 degrees. These angles have the same trigonometric values.

How do you determine the sign of trigonometric functions in different quadrants?

The sign of trigonometric functions depends on the quadrant in which the angle lies. In quadrant 1, all trigonometric functions (sine, cosine, tangent) are positive. In quadrant 2, sine is positive while cosine and tangent are negative. In quadrant 3, tangent is positive while sine and cosine are negative. In quadrant 4, cosine is positive while sine and tangent are negative. This can be remembered using the mnemonic ASTC (All Students Take Calculus).

How do you find the reference angle for an angle in radians?

To find the reference angle for an angle in radians, you measure the angle to the nearest x-axis. For example, if you have an angle of 5π/6, you subtract it from π (the nearest x-axis), resulting in a reference angle of π/6. If the angle is in quadrant 3, such as 7π/6, you subtract π from the angle, giving you a reference angle of π/6. For angles in quadrant 4, like 11π/6, you subtract the angle from 2π, resulting in a reference angle of π/6.