Question Video: Using the Unit Circle to Express the Values of Sine, Cosine, and Tangent for 2𝜋 − 𝑥 in Terms of Their Values for 𝑥, Where 𝑥 Is Any Real Number | Nagwa Question Video: Using the Unit Circle to Express the Values of Sine, Cosine, and Tangent for 2𝜋 − 𝑥 in Terms of Their Values for 𝑥, Where 𝑥 Is Any Real Number | Nagwa

# Question Video: Using the Unit Circle to Express the Values of Sine, Cosine, and Tangent for 2๐ โ ๐ฅ in Terms of Their Values for ๐ฅ, Where ๐ฅ Is Any Real Number Mathematics

In the figure, points ๐(cos ๐, sin ๐) and ๐ lie on the unit circle, and โ ๐ด๐๐ = 2๐ โ ๐. Express the values of sine, cosine, and tangent of 2๐ โ ๐ in terms of their values for ๐. Check whether this is valid for all values of ๐.

06:14

### Video Transcript

In the figure, points ๐ cos ๐, sin ๐ and ๐ lie on the unit circle, and angle ๐ด๐๐ is equal to two ๐ minus ๐. Express the values of sin, cos, and tan of two ๐ minus ๐ in terms of their values for ๐. Check whether this is valid for all values of ๐.

We are told in the question that point ๐ has coordinates cos ๐, sin ๐. And from the diagram, we see that the angle ๐ด๐๐ is equal to ๐. We know that this is true for any point that lies on the unit circle, where ๐ is measured in the counterclockwise direction from the positive ๐ฅ-axis. Since the reflex angle ๐ด๐๐ is equal to two ๐ minus ๐, point ๐ has coordinates cos two ๐ minus ๐, sin two ๐ minus ๐. Using the fact that there are two ๐ radians in a full circle and we measure negative angles in a clockwise direction from the positive ๐ฅ-axis, then the coordinates of point ๐ can also be written as cos negative ๐, sin negative ๐.

From the symmetry of the unit circle, points ๐ and ๐ will both have the same ๐ฅ-coordinate. This means that cos of negative ๐ is equal to cos of ๐. This is actually a standard result that we can quote moving forwards. Since the cosine function is even, the cos of negative ๐ is equal to cos ๐. And since the cos of two ๐ minus ๐ is equal to cos of negative ๐, it must also be equal to cos ๐.

The ๐ฅ-coordinate of point ๐ on the unit circle can also be written as cos ๐. When dealing with the ๐ฆ-coordinates of points ๐ and ๐, we see that ๐ is the same distance above the ๐ฅ-axis as point ๐ is below the ๐ฅ-axis. This means that the sin of negative ๐ is equal to negative sin ๐. In the same way as the result for the cos of negative ๐, this result holds for all values of ๐ since sine is an odd function. The sin of negative ๐ is always equal to negative sin ๐. This means that since sin of two ๐ minus ๐ is equal to sin of negative ๐, it is also equal to negative sin ๐. The ๐ฆ-coordinate of point ๐ can be written as negative sin ๐.

We have now expressed the values of sin and cos of two ๐ minus ๐ in terms of their values for ๐. sin of two ๐ minus ๐ is equal to negative sin ๐ and cos of two ๐ minus ๐ is equal to cos ๐. We can now find an expression for the tan of two ๐ minus ๐ using one of our trigonometric identities. We know that the tan of any angle ๐ผ is equal to sin ๐ผ divided by cos ๐ผ. If we divide the first equation by the second, we have the sin of two ๐ minus ๐ over cos of two ๐ minus ๐ is equal to negative sin ๐ over cos ๐. Using the identity below, the left-hand side simplifies to the tan of two ๐ minus ๐ and the right-hand side to negative tan ๐.

We now have the three expressions required, the values of sin, cos, and tan of two ๐ minus ๐ in terms of their values for ๐. We are also asked to check whether this is valid for all values of ๐. If we let point ๐ lie in the first quadrant, as shown in the diagram, where the angle ๐ด๐๐ is equal to some other value of ๐, then the point ๐, where the counterclockwise angle ๐ด๐๐ is equal to two ๐ minus ๐, will lie in the fourth quadrant as shown.

Once again, these points will have the same ๐ฅ-coordinate, whereas the ๐ฆ-coordinates will be the additive inverse of one another. If point ๐ has the coordinates ๐ฅ, ๐ฆ, then point ๐ will have the coordinates ๐ฅ, negative ๐ฆ. And we can therefore conclude that the expressions for the sin, cos, and tan of two ๐ minus ๐ in terms of their values of ๐ are true for all values of ๐ in the unit circle. The sin of two ๐ minus ๐ is equal to negative sin ๐. The cos of two ๐ minus ๐ is equal to cos ๐. And the tan of two ๐ minus ๐ is equal to negative tan ๐. Note that in all three of these, two ๐ radians can be replaced with 360 degrees.