1) Find the exact value. Use a sum or difference identity.
tan (-15 degrees)
2) Rewrite the following expression as a trigonometric function of a single angle measure.
cos 3x cos 4x - sin 3x sine 4x
1.
tan 15 = tan(45-30)
= (tan45-tan30)/(1 + tan45tan30)
= (1 - 1/√3)/(1 + 1/√3)
multiply by √3/√3
= (√3-1)/(√3+1)
but tan(-15) = -tan15
so
tan(-15°) = (1-√3)/(√3 + 1)
cos 3x cos 4x - sin 3x sine 4x
= cos(3x+4x) = cos 7x
1) To find the exact value of tan (-15 degrees), we can use the sum identity for tangent:
tan (A - B) = (tan A - tan B) / (1 + tan A tan B)
In this case, we want to find tan (-15 degrees), so let A = 0 degrees and B = 15 degrees.
Using the sum identity, we have:
tan (-15 degrees) = (tan 0 degrees - tan 15 degrees) / (1 + tan 0 degrees tan 15 degrees)
Since tan 0 degrees is 0, we can simplify it further:
tan (-15 degrees) = (0 - tan 15 degrees) / (1 + 0 * tan 15 degrees)
tan (-15 degrees) = - tan 15 degrees
Therefore, the exact value of tan (-15 degrees) is equal to - tan 15 degrees.
2) To rewrite the expression cos 3x cos 4x - sin 3x sin 4x as a trigonometric function of a single angle measure, we can use the difference identity for cosine:
cos (A - B) = cos A cos B + sin A sin B
In this case, we have cos 3x cos 4x - sin 3x sin 4x. Using the difference identity, we can rewrite it as:
cos (3x - 4x)
Simplifying further, we have:
cos (-x)
Therefore, the given expression can be rewritten as the trigonometric function cos(-x).
1) To find the exact value of tan(-15 degrees) using a sum or difference identity, we can use the identity:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
In this case, we want to find tan(-15 degrees), so A = 0 degrees and B = 15 degrees. Plugging these values into the identity, we have:
tan(0 - 15) = (tan(0) - tan(15)) / (1 + tan(0)tan(15))
Since tan(0) = 0 and tan(15) = 0.2679, we can substitute these values and simplify:
tan(-15) = (0 - 0.2679) / (1 + 0 * 0.2679)
tan(-15) = -0.2679 / 1 = -0.2679
Therefore, the exact value of tan(-15 degrees) is -0.2679.
2) To rewrite the expression cos 3x cos 4x - sin 3x sine 4x as a trigonometric function of a single angle measure, we can use the double-angle identities and the product-to-sum identities.
First, we can use the double-angle identity for cosine:
cos(2A) = 2cos^2(A) - 1
The expression cos 3x cos 4x can be rewritten as:
cos 3x cos 4x = (1/2) * (cos(3x - 4x) + cos(3x + 4x))
Simplifying this further:
cos 3x cos 4x = (1/2) * (cos(-x) + cos(7x))
Next, we can use the product-to-sum identities for sine:
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B
Applying these identities to the expression sin 3x sine 4x:
sin 3x sine 4x = (1/2) * (sin(3x - 4x) - sin(3x + 4x))
Simplifying this further:
sin 3x sine 4x = (1/2) * (sin(-x) - sin(7x))
Now, we can rewrite the entire expression as a trigonometric function of a single angle measure:
cos 3x cos 4x - sin 3x sine 4x
= (1/2) * (cos(-x) + cos(7x)) - (1/2) * (sin(-x) - sin(7x))
= (1/2) * (cos(-x) + cos(7x) - sin(-x) + sin(7x))
Therefore, the expression cos 3x cos 4x - sin 3x sine 4x can be rewritten as (1/2) * (cos(-x) + cos(7x) - sin(-x) + sin(7x)), which is a trigonometric function of a single angle measure.