Final Answer:
(a) To find the exact value of cos(315°) without a calculator, we use the concept of reference angles. The reference angle for 315° is 45°, and since cosine is negative in the fourth quadrant, cos(315°) is equal to -cos(45°).
Step-by-step explanation:
(a) When faced with the trigonometric expression cos(315°) without a calculator, the process involves utilizing the reference angle and quadrant-specific properties. First, identify the reference angle for 315°, which is the positive acute angle between the terminal side of the given angle and the x-axis. In this case, the reference angle is 45°.
(b) Next, recognize that cosine is negative in the fourth quadrant. Therefore, cos(315°) can be expressed as -cos(45°). Now, recall the exact value of cos(45°), which is √2/2. Applying the appropriate sign based on the quadrant (negative for the fourth quadrant), we obtain the final result: cos(315°) = -√2/2.
(c) This step-by-step process showcases the application of reference angles and quadrant-specific rules to find the exact value of the trigonometric expression without a calculator. Understanding these fundamental principles enhances our ability to handle trigonometric problems effectively, promoting a deeper comprehension of mathematical concepts and strengthening problem-solving skills.
Question:
Given the trigonometric expression cos(315°), explain how you would use the concept of reference angles to find the exact value without using a calculator. Provide the step-by-step process, including identifying the reference angle, determining the relevant trigonometric function, and applying the appropriate sign based on the quadrant?