find the value of ¨ª(theta) for 0 ¡ ¨ª(theta) ¡ 2¥ð(pi)
a)sec¨ª = ¡î2
b)tan¨ª = -1
c)sin¨ª = 1/¡î2
d)cot¨ª = ¡î3
To find the value of ¨ª (theta) within the given range, we need to examine the trigonometric functions and their values.
a) sec¨ª = ¡î2
The secant (sec) function is the reciprocal of the cosine (cos) function. To find the value of ¨ª for which sec¨ª is equal to ¡î2, we can use the following steps:
1. Recall that sec¨ª = 1/cos¨ª.
2. Solve the equation 1/cos¨ª = ¡î2 for ¨ª.
To solve the equation, we can take the reciprocal of both sides to get cos¨ª = -1/2. From the unit circle or trigonometric identities, we find this value occurs for ¨ª = 2¥ð/3 or 4¥ð/3 in the given range.
b) tan¨ª = -1
The tangent (tan) function is the ratio of the sine (sin) and cosine (cos) functions. To determine the value of ¨ª for which tan¨ª is equal to -1, we can follow these steps:
1. Recall that tan¨ª = sin¨ª / cos¨ª.
2. Solve the equation sin¨ª / cos¨ª = -1 for ¨ª.
Rearranging the equation, we have sin¨ª = -cos¨ª. From the unit circle or trigonometric identities, we find this value occurs for ¨ª = 3¥ð/4 in the given range.
c) sin¨ª = 1/¡î2
To find the value of ¨ª for which sin¨ª is equal to 1/¡î2, we can use the following steps:
1. Recall that sin¨ª is equal to the y-coordinate on the unit circle.
2. Determine the angle in the given range that has a y-coordinate of 1/¡î2.
From the unit circle or trigonometric identities, we find this value occurs for ¨ª = ¥ð/6 in the given range.
d) cot¨ª = ¡î3
The cotangent (cot) function is the reciprocal of the tangent (tan) function. To determine the value of ¨ª for which cot¨ª is equal to ¡î3, we can follow these steps:
1. Recall that cot¨ª = 1/tan¨ª.
2. Solve the equation 1/tan¨ª = ¡î3 for ¨ª.
Taking the reciprocal of both sides, we have tan¨ª = -1/¡î3. From the previous explanation, we found that tan¨ª = -1 occurs for ¨ª = 3¥ð/4. Therefore, multiplying both the numerator and denominator by ¡î3 gives tan¨ª = -1/¡î3, corresponding to ¨ª = 3¥ð/4 in the given range.
Therefore, the value of ¨ª that satisfies each equation within the given range is:
a) ¨ª = 2¥ð/3 or 4¥ð/3
b) ¨ª = 3¥ð/4
c) ¨ª = ¥ð/6
d) ¨ª = 3¥ð/4