How to Find the Measure of an Angle When Given the Secant
In trigonometry, the secant is identified as the ratio of the length of the hypotenuse of a right triangle to the length of the leg adjacent to the specified angle. This is often written in shorthand as: sec(x) = (h / a). The secant function is also found to be the reciprocal of the cosine function, since cos(x) = (a / h). It is this relationship between secant and cosine that is necessary in determining an angle when given its secant.
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Instructions
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Use the given value for sec(x) of a triangle to calculate cos(x). A reciprocal is the "flipped over" version of a number. For example, if you are given sec(x) = (5 / 2) then cos(x) = (2 / 5).
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Use a calculator to calculate the value for the inverse cosine of x. For example, if cos(2 / 5) then arccos(2 / 5) = 66.4 degrees.
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Solve for sine to confirm your answer. Since you know secant and therefore cosine, you know the length of both the hypotenuse and the adjacent leg. Use the Pythagorean Theorem a^2 + b^2 = c^2 to solve for the other leg in order to find sine. For example, if the cosine of the angle is known to be (2 / 5) then the adjacent leg is length 2 and the hypotenuse is length 5. Using the Pythagorean Theorem finds:
(2^2 + b^2 = 5^2). Solving for b:( b^2 = 5^2 - 2^2); b = sqrt(5^2 - 2^2) = 4.582.
Since sine is defined as opposite / hypotenuse sine becomes (4.582 / 5). Finding arcsin(4.582 / 5) = 66.4, confirming your result.
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