Find the Exact Value cos(-495 degrees )
The problem presented is asking for the calculation of the cosine of an angle measured in degrees, specifically for the negative angle of -495 degrees. The use of the trigonometric function cosine is required here to determine the ratio of the adjacent side to the hypotenuse in a right-angled triangle corresponding to the given angle, even though the actual context of a triangle is not necessary due to the periodic nature of the cosine function. The question also specifies that the exact value is desired, which implies that the answer should not be an approximation but rather an exact number, likely expressed in terms of square roots or well-known trigonometric values. Additionally, the negative angle suggests that knowledge of the cosine function's symmetry and periodic properties will be important in solving this problem.
$cos \left(\right. - 495 ° \left.\right)$
Answer
Solution:
Step 1:
Normalize the angle by adding or subtracting multiples of $360$掳 until it lies within the range of $0$掳 to $360$掳. For $-495$掳, add $360$掳 once to get $-495 + 360 = -135$掳.
Step 2:
Identify the reference angle and determine the sign of the cosine function based on the quadrant. Since $-135$掳 is in the second quadrant where cosine is negative, the reference angle is $180 - (-135) = 315$掳, and we use a negative sign: $- \cos(315\text{掳})$.
Step 3:
Compute the cosine of the reference angle. The cosine of $315$掳, which is the same as $45$掳, is known to be $\frac{\sqrt{2}}{2}$.
Step 4:
Combine the sign from step 2 with the value from step 3 to get the final answer. The exact value of $\cos(-495\text{掳})$ is $-\frac{\sqrt{2}}{2}$. In decimal form, it is approximately $-0.70710678$.
Knowledge Notes:
The process of finding the exact value of a cosine function for any given angle involves several key knowledge points:
Coterminal Angles: Angles that differ by full rotations (multiples of $360$掳 or $2\pi$ radians) are coterminal and have the same trigonometric values.
Reference Angles: The reference angle is the acute angle that a given angle makes with the x-axis. It is always between $0$掳 and $90$掳 and is used to find the trigonometric values of angles in any quadrant.
Trigonometric Functions in Different Quadrants: The sign of the cosine function depends on the quadrant in which the terminal side of the angle lies. Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
Exact Values of Trigonometric Functions: Certain angles have known exact values for their sine, cosine, and tangent functions. For example, the cosine of $45$掳 is $\frac{\sqrt{2}}{2}$.
Conversion between Degrees and Radians: Trigonometric functions can be evaluated in both degrees and radians. It is important to use the correct unit or convert between them when necessary.
By understanding and applying these concepts, one can solve trigonometric problems involving angles of any measure.
