<\/span><\/h2>\nTo find the exact value of sec, follow these steps:<\/p>\n
Step 1:<\/b> Determine the given angle, usually in degrees or radians.<\/p>\n
Step 2:<\/b> Recall the reciprocal relationship between the secant and cosine functions: sec(\u03b8) = 1\/cos(\u03b8).<\/p>\n
Step 3:<\/b> Identify the reference angle for the given angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.<\/p>\n
Step 4:<\/b> Determine the sign of sec based on the quadrant in which the angle lies. Remember that sec is positive in the first and fourth quadrants, and negative in the second and third quadrants.<\/p>\n
Step 5:<\/b> Use a unit circle or trigonometric identities to find the exact value of cos(\u03b8).<\/p>\n
Step 6:<\/b> Apply the reciprocal relationship to find the exact value of sec(\u03b8) by taking the reciprocal of the cosine value obtained in step 5.<\/p>\n
Step 7:<\/b> Consider any additional restrictions or ranges of values based on the context of the problem you are solving.<\/p>\n
Example:<\/b> Let’s find the exact value of sec(\u03c0\/4).<\/p>\n
Step 1:<\/b> The given angle is \u03c0\/4.<\/p>\n
Step 2:<\/b> sec(\u03c0\/4) = 1\/cos(\u03c0\/4).<\/p>\n
Step 3:<\/b> The reference angle for \u03c0\/4 is also \u03c0\/4 since it already lies in the first quadrant.<\/p>\n
Step 4:<\/b> In the first quadrant, sec is always positive.<\/p>\n
Step 5:<\/b> cos(\u03c0\/4) = \u221a2\/2 (using the properties of a 45-45-90 triangle or unit circle).<\/p>\n
Step 6:<\/b> sec(\u03c0\/4) = 1\/(\u221a2\/2) = 2\/\u221a2 = \u221a2.<\/p>\n
Therefore, the exact value of sec(\u03c0\/4) is \u221a2.<\/p>\n