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formed by the line OA and the x axis. These coordinates are called polar coordinates, and they are written in the form A = (r, θ ). (Note that the polar coordinates of a point are not unique.) In general, for any angle θ, we define the values of sin θ and cos θ as the coordinates of points on the unit circle. Indeed, for any θ, there is a unique point A = (x0, y0) (in rectangular coordinates) on the unit circle ω such that A = (1,θ) (in polar coordinates). We define cos θ = x0 and sin θ = y0; that is, A = (cos θ ,sin θ ) if and only if A = (1,θ) in polar coordinates. From the definition of the sine and cosine functions, it is clear that for all integers k, sin(θ + k · 360◦) = sin θ and cos(θ + k · 360◦) = cos θ; that is, they are periodic functions with period 360◦. For θ = (2k + 1) · 90◦, we define tan θ = sin θ cos θ ; and for θ = k · 180◦, we define cot θ = cos θ sin θ . It is not difficult to see that tan θ is equal to the slope of a line that forms a standard angle of θ with the x axis. A A B C1 C2 D E x x y y O O Figure 1.11. Assume that A = (cos θ ,sin θ ). Let B be the point on ω diametrically opposite to A. Then B = (1, θ + 180◦) = (1, θ − 180◦). Because A and B are symmetric with respect to the origin, B = (− cos θ , − sin θ ). Thus sin(θ ± 180◦) = − sin θ , cos(θ ± 180◦) = − cos θ. It is then easy to see that both tan θ and cot θ are functions with a period of 180◦. Similarly, by rotating point A around the origin 90◦ in the counterclockwise direction (to point C2 in Figure 1.11), in the clockwise direction (to C1), reflecting across the x axis (to D), and reflecting across the y axis (to E), we can show that sin(θ + 90◦) = cos θ , cos(θ + 90◦) = − sin θ , sin(θ − 90◦) = − cos θ , cos(θ − 90◦) = sin θ , sin(−θ ) = − sin θ , cos(−θ ) = cos θ , sin(180◦ − θ ) = sin θ , cos(180◦ − θ ) = − cos θ. 12 103 Trigonometry Problems Furthermore, by either reflecting A across the line y = x or using the second and third formulas above, we can show that sin(90◦ − θ ) = cos θ and cos(90◦ − θ ) = sin θ. This is the reason behind the nomenclature of the "cosine" function: "cosine" is the complement of sine, because the angles 90◦−θ and θ are complementary angles. All these interesting and important trigonometric identities are based on the geometric properties of the unit circle. Earlier, we found addition and subtraction formulas defined for angles α and β with 0◦ < α, β < 90◦ and α +

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