Trigonometry Question
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Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following statements best explains the symmetry of cos(Θ)? Assume that Θ can be interpreted as an angle in standard position, whose terminal arm intersects circle A at the point (X1,Y1), where X1,Y1 are real numbers.
- Since we know that sine and cosine are the same function, except that cosine is different by a phase shift of π2, therefore the same properties must apply to both functions. Thus, cosine is an odd function.
- Using the same reasoning as was applied to the sine function in the previous questions, it can be seen that that cos(Θ)=X1 and cos(-Θ)=-X1. Therefore cos(Θ)=-cos(-Θ), showing that cosine is odd.
- For any given angle Θ, the value of cos(Θ) can be interpreted as X1. Using similar reasoning as was used in previous questions, it can be seen that cos(-Θ)=X1, where X1 will always be of the same sign. Hence, cos(Θ)=cos(-Θ), showing that the cosine function is even.
- Looking at the unit circle, the sine of any angle Θ will be Y1 and the cosine of the angle will be X1. The properties of the sine function can therefore be seen in the properties of the cosine function, if we simply reflect the coordinate axis about the line y=x. Therefore, an odd function will become an even function, and thus cosine is an even function.