60 degrees

Definition

60 degrees is a specific angle measure that is significant in trigonometry, especially in the study of triangles and circular functions. It is one of the angles in an equilateral triangle, where all angles are equal, and each measures 60 degrees. Understanding 60 degrees is crucial for calculating sine, cosine, and tangent values, as well as for applying sum and difference identities in trigonometric equations.

5 Must Know Facts For Your Next Test

1. The sine of 60 degrees is $$\frac{\sqrt{3}}{2}$$, which is important for solving various trigonometric problems.
2. The cosine of 60 degrees equals $$\frac{1}{2}$$, which is often used in calculations involving angles in the unit circle.
3. In terms of radians, 60 degrees is equivalent to $$\frac{\pi}{3}$$ radians.
4. The tangent of 60 degrees equals $$\sqrt{3}$$, which is significant when determining slope or angle in practical applications.
5. In sum and difference identities, 60 degrees is often used with angles like 30 degrees to derive other trigonometric values.

Review Questions

• How does the measure of 60 degrees relate to properties of an equilateral triangle?
  • In an equilateral triangle, each angle measures exactly 60 degrees, which means that all sides are also equal. This relationship between the angles and sides creates a fundamental property that is crucial for understanding trigonometric ratios. When solving problems related to triangles, knowing that each angle in an equilateral triangle is 60 degrees helps establish relationships for calculating side lengths using sine, cosine, and tangent.
• Demonstrate how you would use the sine and cosine values for 60 degrees in a practical trigonometric problem.
  • When solving a problem involving a right triangle where one angle is 60 degrees, you can use the sine and cosine values to find unknown sides. For instance, if you have a hypotenuse of length 10 units, the opposite side can be calculated using sine: $$\text{opposite} = \sin(60) \cdot 10 = \frac{\sqrt{3}}{2} \cdot 10 = 5\sqrt{3}$$. The adjacent side can be calculated with cosine: $$\text{adjacent} = \cos(60) \cdot 10 = \frac{1}{2} \cdot 10 = 5$$. These calculations demonstrate how critical the values for 60 degrees are in determining side lengths.
• Evaluate the significance of 60 degrees in deriving sum and difference identities in trigonometry.
  • The angle of 60 degrees plays a pivotal role in deriving sum and difference identities. For example, when using the identity for $$\sin(A + B)$$, if A is 60 degrees and B is another angle like 30 degrees, you can express it as $$\sin(60 + 30) = \sin(60)\cos(30) + \cos(60)\sin(30)$$. Plugging in known values results in specific numerical outcomes that help solve complex problems. Thus, the presence of 60 degrees allows for simplification and derivation of various trigonometric identities, enhancing problem-solving strategies.

Related terms

Equilateral Triangle: A triangle where all three sides are of equal length and each angle measures 60 degrees.

Sine Function: A trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

Cosine Function: A trigonometric function that represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.