suppose beta is an angle in the second quadrant and tan beta=-2. Fine the exact vaule of sin beta and cos beta
draw a right-angled triangle in the second quad
so that tan B = -3/1, that is make the opposite 3 and the adjacent -1, thus the hypotenuse is √10
so y=3, x = -1, r=√10
sinB = 3/√10, cosB = -1/√10
To find the exact values of sine (sin) and cosine (cosine) of an angle β in the second quadrant, where tan(β) = -2, we can use the following steps:
Step 1: Start with the given information that tan(β) = -2.
Step 2: Recall that tangent (tan) is the ratio of the sine (sin) and cosine (cos) of an angle. Therefore, we can set up an equation using the trigonometric identity: tan(β) = sin(β) / cos(β).
Step 3: Substitute the given value of tan(β) = -2 into the equation: -2 = sin(β) / cos(β).
Step 4: Multiply both sides of the equation by cos(β) to isolate sin(β): -2 * cos(β) = sin(β).
Step 5: Next, we need to find the exact value of cos(β). In the second quadrant, the cosine is negative.
Step 6: To determine the exact value of cos(β), we can use the Pythagorean identity: cos²(β) + sin²(β) = 1. Since we already have sin(β) = -2cos(β), we can substitute this value into the identity.
Substituting -2cos(β) for sin(β) in the Pythagorean identity, we get:
cos²(β) + (-2cos(β))² = 1.
Simplifying, we have:
cos²(β) + 4cos²(β) = 1.
5cos²(β) = 1.
cos²(β) = 1/5.
Step 7: Taking the square root of both sides, we get cos(β) = ±√(1/5). In the second quadrant, the cosine is negative, so we take the negative square root: cos(β) = -√(1/5).
Step 8: Now that we have the value of cos(β), we can substitute it back into our equation from Step 4: -2 * (-√(1/5)) = sin(β).
Step 9: Simplifying, we have: 2√(1/5) = sin(β).
Therefore, the exact value of sin(β) is 2√(1/5) and the exact value of cos(β) is -√(1/5).