# How to find a sine if cosine is known?

When a problem is given in which one trigonometric function is known, and it is required to find another trigonometric function, it is not difficult to solve it. But it is very important to take into account the small subtleties in the decision. Consider detailed solutions, given the nuances. There are several variants of problems in which it is required to find a sine, if cosine is known.

Option 1. Given a right triangle. The value of the cosine of the angle of this triangle (not a right angle) is known. Nasty sinus

Decision:

Recall the basic trigonometric identity: sin2α + cos2α =1.

Hence the sin2α = 1 - cos2α.

sin α = ± √ (1- cos2α)

In a right triangle, the value of the angle (not right) can be in the range of 10up to 890. The sine of such an angle is always positive, therefore, we will have a plus in front of the root.

Option 2. The cosine of a certain angle is known. It is also known to which quarter of the trigonometric circle the angle belongs.

Decision:

sin2α + cos2α =1.

sin2α = 1 - cos2α.

sin α = ± √ (1- cos2α)

It is known that the trigonometric function sine can take values from -1 to +1. Therefore, removing the root, we must take this into account. Depending on which quarter the angle belongs to, put a sign in front of the root “+” or “-”.

What are the quarters:

- I (first) - α from 00up to 900;
- II (second) - α from 900up to 1800;
- III (third) - α from 1800up to 2700;
- IV (fourth) - α from 2700up to 3600.

If the angle belongs to the I or II quarter, then we do not set the sign of the root "-", since in this case sin α is always positive.

If the angle belongs to the third or fourth quarter, then we put a “-” in front of the root sign, since in this case sin α is always negative.

Example. Given the cosine, find the sine. cos α = v3 / 2. Angle in the fourth quarter.

Decision:

So, how to find the sine, knowing the cosine:

sin α = ± v (1- cos2α)

Since, by the condition of the problem, the angle belongs to the fourth quarter of the trigonometric circle, we put the sign “-” in front of the root.

sin α = - v (1-3 / 4)

sin α = - 1/2.

Example. In a right triangle the cosine of one angle is 1/2. Find the sine of this angle.

Solution: sin2α + cos2α =1.

sin2α = 1 - cos2α.

Since we are looking for the angle of a right triangle, in front of the root is the “+” sign.

sin α = v (1- cos2α)

sin α = v (1-1 / 4)

sin α = v3 / 2.

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