Simplifying steps without using the calculator for:
tan(cos^-1(-1/10))
cos(sin^−1(1/x)) Assume x is positive
tan(cos^−1(12/13))
cos^−1(cos 150°)
This is pretty much the entire section we are doing. My teacher is a robot and has us self teach ourselves. Can someone please tell me how I am supposed to do these without a calculator???
last one: the angle whose cosine is cosine of 150? You have to be kidding..150.
next to last one:
tan theta=sinTheta/cosTheta where theta is arccos12/13
hmmm then you have
sin arcsos( )/cos arc cos( ) or
sin arecos ( ). Now draw the triangle, isn't it a a 5 12 13 triangle. so the sin is 5/13? check that.
the second one. draw the triangle. I see the last side as sqrt (x^2 -1). so cos must be sqrt(1-x^2)/x
So, how are you supposed to do these without a calculator? Analyze the expressions.
I can definitely help you understand how to solve these problems without using a calculator. Let's break down each problem step by step:
1. tan(cos^-1(-1/10)):
- Start by finding the value of cos^-1(-1/10). This means finding the angle whose cosine equals -1/10.
- Since cosine is negative in the second and third quadrants, we need to find the angle between 90° and 270° that has a cosine of -1/10.
- To find this angle, you can use the identity: cos^-1(-x) = π - cos^-1(x)
- So, cos^-1(-1/10) = π - cos^-1(1/10)
- Now, find the value of cos^-1(1/10). This means finding the angle whose cosine equals 1/10.
- Using the inverse cosine function, you can find this angle as approximately 84.3°.
- Therefore, cos^-1(-1/10) ≈ π - 84.3° ≈ 95.7°
- Finally, find the tangent of 95.7° to get the answer.
2. cos(sin^−1(1/x)) (assuming x is positive):
- Start by finding the value of sin^−1(1/x). This means finding the angle whose sine equals 1/x.
- Using the inverse sine function, you can find this angle as sin^−1(1/x).
- Now, take the cosine of sin^−1(1/x) to get the answer.
3. tan(cos^−1(12/13)):
- Start by finding the value of cos^−1(12/13). This means finding the angle whose cosine equals 12/13.
- Using the inverse cosine function, you can find this angle as approximately 22.6°.
- Finally, find the tangent of 22.6° to get the answer.
4. cos^−1(cos 150°):
- Since the inverse cosine function gives angles between 0° and 180°, we need to find an angle between 0° and 180° whose cosine is the same as cos 150°.
- Using the unit circle or the periodicity of cosine, we can determine that cos 150° is equivalent to cos 30°.
- Therefore, cos^−1(cos 150°) = 30°.
Remember, these steps are providing a general approach to solving these problems without a calculator. It's important to have a strong understanding of trigonometric concepts, identities, and properties to solve such problems accurately.