D. 30° Degree Sin 30° = Cos 30° = Tan 30° = E. 60° Degree Sin 60° = Cos 60° = Tan 60°
The Correct Answer and Explanation is:
Here are the correct answers for the trigonometric functions listed in the image.
D. 30° Degree
Sin 30° = 1/2
Cos 30° = √3/2
Tan 30° = √3/3 (or 1/√3)
E. 60° Degree
Sin 60° = √3/2
Cos 60° = 1/2
Tan 60° = √3
Explanation
The values for these common trigonometric functions are derived from the geometric properties of a special right triangle, known as the 30-60-90 triangle.
To create this triangle, we can begin with an equilateral triangle, which has three equal sides and three equal angles of 60 degrees. Let’s imagine this triangle has sides of length 2. If we draw an altitude (a perpendicular line) from one corner to the opposite base, we split the equilateral triangle into two identical right-angled triangles.
Each of these new triangles has angles measuring 30, 60, and 90 degrees. The hypotenuse (the side opposite the 90-degree angle) is the original side of the equilateral triangle, so its length is 2. The shortest side is opposite the 30-degree angle, and its length is half the base of the original equilateral triangle, making it 1. The third side, which is the altitude we drew, can be found using the Pythagorean theorem (a² + b² = c²). This calculation shows its length to be the square root of 3 (√3).
With the side lengths of our 30-60-90 triangle established as 1, √3, and 2, we can apply the basic trigonometric ratios known by the mnemonic SOH CAH TOA:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
For the 30-degree angle, the opposite side is 1, the adjacent side is √3, and the hypotenuse is 2. This gives us:
- Sin 30° = 1/2
- Cos 30° = √3/2
- Tan 30° = 1/√3, which is commonly written with a rationalized denominator as √3/3.
For the 60-degree angle, the opposite side is √3, the adjacent side is 1, and the hypotenuse is 2. This gives us:
- Sin 60° = √3/2
- Cos 60° = 1/2
- Tan 60° = √3/1 = √3
