I am really struggling with how to do these problems, I posted them a few minutes ago but the answers/work shown was incorrect.
1)
a) 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°
b) And Find its exact value.
2)
a)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
b) 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
You must make an effort to memorize the formulas.
(I suggest you memorize the pattern)
cos(A+B) = cosAcosB - sinAsinB
Notice you have that pattern in
cos 12° cos 18° − sin 12° sin 18°
= co(12+18)
= cos 30
which you should have memorized as √3/2 from the 30-6-90 triangle
2) same suggestion, you must know these formulas
sin(A-B) = sinAcosB - cosAsinB
then sin 4pi/5 cos 7pi/15 - cos 4pi/5 sin 7pi/15
= sin(4π/5 - 7π/15)
= sin((1/3)π) or sin π/3 or sin 60°
again you should know that one
sin π/3 = √3/2
3
tanØ/cosØ - secØ
= (sinØ/cosØ) / cosØ - 1/cosØ
= sinØ/cos^2 Ø - cosØ/cos^2 Ø
= (sinØ - cosØ)/cos^2 Ø
Sure! I can help you with these problems step-by-step. Let's tackle them one by one.
1)
a) To write the expression "cos 12° cos 18° - sin 12° sin 18°" 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.
Using this formula, we rewrite the expression as:
cos 12° cos 18° - sin 12° sin 18° = cos (12° - 18°)
b) To find the exact value, we simplify the expression further:
cos (12° - 18°) = cos (-6°) (since 12° - 18° = -6°)
We know that the cosine function has a period of 360°, so we can add 360° to the angle without changing the value:
cos (-6°) = cos (360° - 6°)
Using the symmetry property of the cosine function (cos(-θ) = cos θ), we have:
cos (360° - 6°) = cos 6°
Therefore, the exact value of the expression is cos 6°.
2)
a) Following the same approach, we'll use the subtraction formula for sine: sin(a - b) = sin a cos b - cos a sin b.
Using this formula, we rewrite the expression as:
sin 4π/5 cos 7π/15 - cos 4π/5 sin 7π/15 = sin (4π/5 - 7π/15)
b) To find the exact value, we simplify the expression further:
sin (4π/5 - 7π/15) = sin (12π/15 - 7π/15) (since 4π/5 = 12π/15)
Simplifying the angle, we get:
sin (12π/15 - 7π/15) = sin (5π/15)
We can simplify the angle further:
sin (5π/15) = sin (π/3) (since 5π/15 = π/3)
Using the exact value of sine for π/3, which is sqrt(3)/2, the exact value of the expression is sqrt(3)/2.
3)
To write the expression "tan θ / cos θ - sec θ" in terms of sine and cosine and simplify, we'll start by expressing sec θ in terms of cosine: sec θ = 1/cos θ.
Substituting this in the expression, we have:
tan θ / cos θ - sec θ = tan θ / cos θ - 1/cos θ
To combine the fractions, we need a common denominator, which is cos θ in this case.
Rewriting the expression with a common denominator, we get:
(tan θ - 1) / cos θ
This is the simplified expression in terms of sine and cosine: (tan θ - 1) / cos θ.
I hope this helps! Let me know if you have any further questions.
No problem! I can help you with these trigonometry problems.
Let's start with problem 1:
a) To write the expression as a trigonometric function of one number, we can use the subtraction formula for cosine:
cos(α - β) = cos α cos β + sin α sin β
In this case, α = 12° and β = 18°. Substituting these values into the formula, we get:
cos 12° cos 18° - sin 12° sin 18° = cos(12° - 18°)
b) To find the exact value, we need to evaluate the cosine of the angle 12° - 18°. Simplifying further:
cos(12° - 18°) = cos(-6°)
Note that cosine is an even function, meaning that cos(-θ) = cos(θ). So,
cos(-6°) = cos(6°)
Now, we need to find the value of cos(6°). You can use a calculator or trigonometric table to find the exact value, but let's demonstrate how to calculate it using a series expansion known as Maclaurin series:
cos x = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
For x = 6°, the series expansion becomes:
cos 6° ≈ 1 - (6°)^2/2! + (6°)^4/4! - (6°)^6/6! + ...
Simplifying:
cos 6° ≈ 1 - (36°^2)/(2 * 1) + (1296°^4)/(4 * 3 * 2 * 1) - ...
You can continue this series expansion to get a more accurate approximation, or you can use a calculator to find the value of cos 6°.
Now let's move on to problem 2:
a) Here, we can use the subtraction formula for sine:
sin(α - β) = sin α cos β - cos α sin β
Using this formula, we can rewrite the expression as:
sin 4π/5 cos 7π/15 - cos 4π/5 sin 7π/15 = sin(4π/5 - 7π/15)
b) To find the exact value, we need to evaluate the sine of the angle 4π/5 - 7π/15. Simplifying further:
sin(4π/5 - 7π/15) = sin(12π/15 - 7π/15)
sin(12π/15 - 7π/15) = sin(5π/15)
Since 5π/15 can be simplified to π/3, the exact value of sin(5π/15) is sin(π/3). You can use a calculator or trigonometric table to find the exact value of sin(π/3).
Finally, let's move on to problem 3:
To write the trigonometric expression tan θ / cos θ - sec θ in terms of sine and cosine, we need to use the relationships:
tan θ = sin θ / cos θ
sec θ = 1 / cos θ
Substituting these relationships into the expression, we get:
tan θ / cos θ - sec θ = (sin θ / cos θ) / cos θ - 1 / cos θ
Now, we can combine the fractions by finding a common denominator:
[(sin θ) - 1] / cos θ
So, the simplified trigonometric expression in terms of sine and cosine is (sin θ - 1) / cos θ.
I hope this helps! Let me know if you have any more questions.