1)
Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number.
cos 12° cos 18° − sin 12° sin 18°
And Find its exact value.
2)
Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number.
sin 4pi/5 cos 7pi/15 - cos 4pi/5 sin 7pi/15
And find its exact value.
3)
Write the trigonometric expression in terms of sine and cosine, and then simplify.
tan θ / cos θ − sec θ
1) that would be cos(12°+18°) which I'm sure you know
2) sin(4π/5-7π/5)
3)
well,
tan/cos = sin/cos^2 and sec = 1/cos
Thank you so much but are those the expressions or exact answers for the first two problems?
they are expressions. You surely know the exact value of cos(30°)
as for sin(-3π/5) you would have to construct a pentagon and figure out some side lengths. figure out sin 36° and then sin 72° and then the sine of 108°.
1) To write the expression as a trigonometric function of one number, we can use the subtraction formula for cosine:
cos (A - B) = cos A cos B + sin A sin B
In this case, A = 18° and B = 12°. So the expression becomes:
cos (18° - 12°) = cos (6°)
To find the exact value of cos 6°, you can use a calculator or a trigonometric table.
2) Similarly, to write the expression as a trigonometric function of one number, we can use the subtraction formula for sine:
sin (A - B) = sin A cos B - cos A sin B
In this case, A = 4π/5 and B = 7π/15. So the expression becomes:
sin (4π/5 - 7π/15) = sin (12π/15 - 7π/15) = sin (5π/15) = sin (π/3)
To find the exact value of sin π/3, which is a well-known value, you can use a calculator or a trigonometric table.
3) To write the expression in terms of sine and cosine, we need to express tan θ, cos θ, and sec θ in terms of sine and cosine:
tan θ = sin θ / cos θ
sec θ = 1 / cos θ
So the expression becomes:
tan θ / cos θ − sec θ = (sin θ / cos θ) / cos θ − 1 / cos θ
To simplify this expression, we need to combine the fractions over a common denominator:
[(sin θ) - 1] / cos θ
Therefore, the simplified expression in terms of sine and cosine is (sin θ - 1) / cos θ.