I need help finding the value of the trig functions
8. Use special right triangles to state the value of the 6 trig functions for 30 degrees, 45 degrees and 60 degrees
for those three triangles, your text will have a table with all the desired values.
for example,
sin 30 = 1/2
tan 45 = 1
and so on
To find the values of the trigonometric functions for 30 degrees, 45 degrees, and 60 degrees, we can use special right triangles. Specifically, we will use the 30-60-90 triangle and the 45-45-90 triangle.
1. 30-degree angle:
In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. Let's assume that the hypotenuse (the side opposite the 90-degree angle) has a length of 2 (you can choose any value as long as you keep the ratio consistent). Based on this assumption, the other two sides are 1 (opposite the 30-degree angle) and √3 (opposite the 60-degree angle).
The trigonometric functions for 30 degrees are:
- Sine (sin): opposite/hypotenuse = 1/2
- Cosine (cos): adjacent/hypotenuse = √3/2
- Tangent (tan): opposite/adjacent = 1/√3 = √3/3
- Cosecant (csc): 2/1 = 2
- Secant (sec): 2/√3
- Cotangent (cot): √3
2. 45-degree angle:
In a 45-45-90 triangle, the sides are in the ratio 1:1:√2. Assuming that the legs (the two sides opposite the 45-degree angles) have a length of 1 (again, you can choose any value as long as you keep the ratio consistent), the hypotenuse will have a length of √2.
The trigonometric functions for 45 degrees are:
- Sine (sin): opposite/hypotenuse = 1/√2 = √2/2
- Cosine (cos): adjacent/hypotenuse = 1/√2 = √2/2
- Tangent (tan): opposite/adjacent = 1/1 = 1
- Cosecant (csc): √2
- Secant (sec): √2
- Cotangent (cot): 1
3. 60-degree angle:
Using the 30-60-90 triangle, assuming the hypotenuse has a length of 2, the opposite side (opposite the 60-degree angle) will be √3.
The trigonometric functions for 60 degrees are:
- Sine (sin): opposite/hypotenuse = √3/2
- Cosine (cos): adjacent/hypotenuse = 1/2
- Tangent (tan): opposite/adjacent = √3/1 = √3
- Cosecant (csc): 2/√3
- Secant (sec): 2
- Cotangent (cot): 1/√3 = √3/3
These are the values of the six trigonometric functions for 30 degrees, 45 degrees, and 60 degrees using special right triangles.