For what value of x is sin(x) equal to cos(19) where 0 ≤ x ≤ 90?

To find the value of x such that sin(x) = cos(19) within the interval 0 ≤ x ≤ 90, we can use the trigonometric identity that states:

• sin(x) = cos(90 – x)

Given this identity, we can rewrite the equation:

sin(x) = cos(19)  
=> cos(90 - x) = cos(19)

From the equation cos(A) = cos(B), we know there are two possible solutions:

• A = B
• A = 360 – B

Applying this to our situation, we have:

• 90 – x = 19
• 90 – x = 360 – 19

Now, let’s solve for x in each case:

Case 1: 90 – x = 19

90 - x = 19  
=> x = 90 - 19  
=> x = 71

Case 2: 90 – x = 341

90 - x = 341  
=> x = 90 - 341  
=> x = -251

Since -251 is not within the defined interval of 0 ≤ x ≤ 90, we discard this solution.

Thus the only solution is:

Final Answer:

x = 71°