Pre Calculus
why is sin(-20)/cos 380
equal to tan (-200)
To understand why sin(-20)/cos(380) is equal to tan(-200), we need to use the identities in trigonometry.
1. First, let's understand that sin(-20) is equal to -sin(20) because the sine function is an odd function.
2. Similarly, cos(380) is equal to cos(380-360) since cos(x) is a periodic function with a period of 360 degrees. Therefore, cos(380) is equal to cos(20).
3. Now, we can rewrite sin(-20)/cos(380) as (-sin(20))/cos(20).
4. According to the quotient identity of the tangent function, tan(x) = sin(x)/cos(x).
5. Therefore, (-sin(20))/cos(20) can be rewritten as tan(-20).
6. Since -200 is congruent to -20 mod 180 degrees and tangent is a periodic function with a period of 180 degrees, tan(-200) is equal to tan(-20).
7. Hence, sin(-20)/cos(380) is equal to tan(-200).
To understand why sin(-20)/cos(380) is equal to tan(-200), we need to use the properties of trigonometric functions.
First, let's simplify the given expression sin(-20)/cos(380):
Using the fact that sin(-x) = -sin(x) and cos(-x) = cos(x), we can rewrite sin(-20) as -sin(20) and cos(380) as cos(-380):
sin(-20)/cos(380) = -sin(20)/cos(-380)
Next, let's look at the relationship between tangent (tan), sine (sin), and cosine (cos). The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle:
tan(x) = sin(x)/cos(x)
Now, we can substitute -200 degrees into the equation:
tan(-200) = sin(-200)/cos(-200)
Since -sin(20) is equivalent to sin(-200) and cos(-380) is equivalent to cos(-200). We can rewrite the equation:
-sin(20)/cos(-380) = sin(-200)/cos(-200)
So, sin(-20)/cos(380) = tan(-200)
I covered this previously.
Sin(theta)=sin(theta+360) in degrees
Sin(-theta)=sin(Theta) in degrees
sin(-20)=-sin(20) =-sin(20+360)=-sin(380)
then...
Tan(-theta)=-tan(theta).
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