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 2𝜋 − 𝑥 in Terms of Their Values for 𝑥, Where 𝑥 Is Any Real Number Mathematics

In the figure, points 𝑀(cos 𝜃, sin 𝜃) and 𝑁 lie on the unit circle, and ∠𝐴𝑂𝑁 = 2𝜋 − 𝜃. Express the values of sine, cosine, and tangent of 2𝜋 − 𝜃 in terms of their values for 𝜃. Check whether this is valid for all values of 𝜃.

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Video Transcript

In the figure, points 𝑀 cos 𝜃, sin 𝜃 and 𝑁 lie on the unit circle, and angle 𝐴𝑂𝑁 is equal to two 𝜋 minus 𝜃. Express the values of sin, cos, and tan of two 𝜋 minus 𝜃 in terms of their values for 𝜃. Check whether this is valid for all values of 𝜃.

We are told in the question that point 𝑀 has coordinates cos 𝜃, sin 𝜃. And from the diagram, we see that the angle 𝐴𝑂𝑀 is equal to 𝜃. We know that this is true for any point that lies on the unit circle, where 𝜃 is measured in the counterclockwise direction from the positive 𝑥-axis. Since the reflex angle 𝐴𝑂𝑁 is equal to two 𝜋 minus 𝜃, point 𝑁 has coordinates cos two 𝜋 minus 𝜃, sin two 𝜋 minus 𝜃. Using the fact that there are two 𝜋 radians in a full circle and we measure negative angles in a clockwise direction from the positive 𝑥-axis, then the coordinates of point 𝑁 can also be written as cos negative 𝜃, sin negative 𝜃.

From the symmetry of the unit circle, points 𝑀 and 𝑁 will both have the same 𝑥-coordinate. This means that cos of negative 𝜃 is equal to cos of 𝜃. This is actually a standard result that we can quote moving forwards. Since the cosine function is even, the cos of negative 𝜃 is equal to cos 𝜃. And since the cos of two 𝜋 minus 𝜃 is equal to cos of negative 𝜃, it must also be equal to cos 𝜃.

The 𝑥-coordinate of point 𝑁 on the unit circle can also be written as cos 𝜃. When dealing with the 𝑦-coordinates of points 𝑀 and 𝑁, we see that 𝑀 is the same distance above the 𝑥-axis as point 𝑁 is below the 𝑥-axis. This means that the sin of negative 𝜃 is equal to negative sin 𝜃. In the same way as the result for the cos of negative 𝜃, this result holds for all values of 𝜃 since sine is an odd function. The sin of negative 𝜃 is always equal to negative sin 𝜃. This means that since sin of two 𝜋 minus 𝜃 is equal to sin of negative 𝜃, it is also equal to negative sin 𝜃. The 𝑦-coordinate of point 𝑁 can be written as negative sin 𝜃.

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

We now have the three expressions required, the values of sin, cos, and tan of two 𝜋 minus 𝜃 in terms of their values for 𝜃. We are also asked to check whether this is valid for all values of 𝜃. If we let point 𝑃 lie in the first quadrant, as shown in the diagram, where the angle 𝐴𝑂𝑃 is equal to some other value of 𝜃, then the point 𝑄, where the counterclockwise angle 𝐴𝑂𝑄 is equal to two 𝜋 minus 𝜃, will lie in the fourth quadrant as shown.

Once again, these points will have the same 𝑥-coordinate, whereas the 𝑦-coordinates will be the additive inverse of one another. If point 𝑃 has the coordinates 𝑥, 𝑦, then point 𝑄 will have the coordinates 𝑥, negative 𝑦. And we can therefore conclude that the expressions for the sin, cos, and tan of two 𝜋 minus 𝜃 in terms of their values of 𝜃 are true for all values of 𝜃 in the unit circle. The sin of two 𝜋 minus 𝜃 is equal to negative sin 𝜃. The cos of two 𝜋 minus 𝜃 is equal to cos 𝜃. And the tan of two 𝜋 minus 𝜃 is equal to negative tan 𝜃. Note that in all three of these, two 𝜋 radians can be replaced with 360 degrees.

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