what is the tangent of pi/5, the cotangent of pi/7, and the co secant of pi/6?
To find the tangent of an angle, you can use a scientific calculator or calculate it manually using trigonometric identities.
1. Tangent of π/5:
- You can use a scientific calculator and directly input the angle π/5 to get the tangent value. It should be approximately 0.7265.
- If you prefer to calculate it manually, the tangent of an angle can be found using the formula: tan(x) = sin(x) / cos(x). Since the angle is π/5, we need to find the values of sin(π/5) and cos(π/5).
- sin(π/5) can be evaluated to approximately 0.5878.
- cos(π/5) can be evaluated to approximately 0.8090.
- Plugging in these values into the tangent formula, we get:
tan(π/5) ≈ sin(π/5) / cos(π/5) = 0.5878 / 0.8090 ≈ 0.7265.
2. Cotangent of π/7:
- Again, you can use a scientific calculator to find the cotangent of π/7 directly. It should be approximately 1.3764.
- If you prefer to calculate it manually, the cotangent of an angle can be found using the formula: cot(x) = cos(x) / sin(x). Since the angle is π/7, we need to find the values of cos(π/7) and sin(π/7).
- cos(π/7) can be evaluated to approximately 0.9009.
- sin(π/7) can be evaluated to approximately 0.4339.
- Plugging in these values into the cotangent formula, we get:
cot(π/7) ≈ cos(π/7) / sin(π/7) = 0.9009 / 0.4339 ≈ 1.3764.
3. Cosecant of π/6:
- The cosecant of an angle is the reciprocal of its sine value. So, to find the cosecant of π/6, we need to find the sine of π/6 and then take its reciprocal.
- The sine of π/6 is 0.5, so the reciprocal of 0.5 is 1/0.5, which equals 2. Therefore, the cosecant of π/6 is 2.
Remember, calculators provide quick and accurate results. However, when calculating manually, it's important to be familiar with trigonometric identities and remember how to evaluate trigonometric functions for common angles.