5.Find the complete exact solution of sin x = -√3/2.
10. Solve cos 2x – 3sin x cos 2x = 0 for the principal value(s) to
two decimal places.
12. Solve tan^2x + tan x – 1 = 0 for the principal value(s) to two
decimal places.
19.Prove that tan^2a – 1 + cos^2a
� = tan^2 a sin^2 a.
29. Prove that tan B� sin B� + cos B� = sec B�.
5.
since sin(pi/3) = √3/2
and sin < 0 in QIII,QIV, you have
x = (2k+1)*pi+pi/3 and 2k*pi-pi/3
10.
cos2x (1 - 3sinx) = 0
cos2x = 0 or sinx = 1/3
cos 2x = 0:
2x = k*pi/2 where k is an integer
x = pi/4, 3pi/4, 5pi/4, 7pi/4
sinx = 1/3:
x = 0.34 or pi-0.34
12.
tanx = 1/2 (-1±√5)
= .618 or -1.618
so take arctan to get x
19.
tan^2-1+cos^2
= sec^2-2+cos^2
= (sec-cos)^2
= (1-cos^2)^2/cos^2
= (sin^2)^2/cos^2
= sin^2/cos^2 * sin^2
= tan^2 sin^2
29.
tan*sin + cos
= sin^2/cos + cos
= (sin^2+cos^2)/cos
= 1/cos
= sec
1.
John Yinger thinks that the natural-born clause should be removed because (1 point)
it makes international policy more difficult.
it goes against a central principle of American democracy.
it damages the relationship between American citizens and immigrant communities.
2.
What do natural-born citizens and naturalized citizens have in common? (1 point)
Both can run for president.
Both are citizens of the United States.
Both are born in the United States.
3.
Which statement shows a contrast between the two opinions in the first selection? (1 point)
The Constitution should be amended.
America is open to foreign-born citizens.
The president should be loyal to our country.
4.
You would expect an authoritarian person to
(1 point)
respect individual freedom.
ask for other people's opinions.
demand total obedience.
5.
When Nicholas Gage says Miss Hurd "nearly dragged" him onto his life's path, he is using (1 point)
comparison.
sequence.
hyperbole.
6.
In "The Teacher Who Changed My Life," the author is trying to persuade readers that
(1 point)
immigrants face lots of difficulties.
schoolteachers are not appreciated enough.
he owes his success to Miss Hurd.
7.
Which of the following sentences contains a compound predicate? (1 point)
The author's mother died to help her children emigrate.
Miss Hurd was strict, yet encouraging.
The author answered the phone and heard his teacher's voice.
8.
Which of the following sentences contains a compound subject?
(1 point)
The author and her sister were dropped at their new school.
The principal was a grim-looking man.
The author did not understand the purpose of hobbies and clubs.
5. To find the complete exact solution of sin x = -√3/2, we can use the unit circle or refer to the trigonometric ratios of special angles. In this case, the value -√3/2 corresponds to the sine of 5𝜋/3 or 150 degrees. Since sine is negative in the third and fourth quadrants, we can write the solution as x = 5𝜋/3 + 2𝜋n or x = 150 degrees + 360 degreesn, where n is an integer.
10. To solve cos 2x – 3sin x cos 2x = 0, let's simplify the equation first. We can factor out cos 2x from both terms: cos 2x(1 - 3sin x) = 0.
To find the values of x that satisfy this equation, we have two cases:
1. cos 2x = 0: Solving this equation gives us two solutions: 2x = 𝜋/2 + 𝜋n or 2x = 3𝜋/2 + 𝜋n, where n is an integer. Dividing by 2, we get x = 𝜋/4 + 𝜋n/2 or x = 3𝜋/4 + 𝜋n/2.
2. 1 - 3sin x = 0: Solving this equation gives us sin x = 1/3. Using inverse sine or referring to trigonometric ratios of special angles, we find two solutions: x = 𝜋/6 + 2𝜋n or x = 5𝜋/6 + 2𝜋n, where n is an integer.
Combining both cases, the principal values of x to two decimal places are x ≈ 0.52, 1.04, 1.74, and 2.62.
12. To solve tan^2x + tan x – 1 = 0, we can use factoring or the quadratic formula. If we factor the equation, we get (tan x - 1)(tan x + 1) = 0. Setting each factor equal to zero gives us two cases:
1. tan x - 1 = 0, which gives us tan x = 1. Using inverse tangent, we find the solution x = 𝜋/4 + 𝜋n, where n is an integer.
2. tan x + 1 = 0, which gives us tan x = -1. Using inverse tangent, we find the solution x = 3𝜋/4 + 𝜋n, where n is an integer.
The principal values of x to two decimal places are x ≈ 0.79 and 2.36.
19. To prove that tan^2a – 1 + cos^2a = tan^2a sin^2a, we can start with the identity sin^2a + cos^2a = 1. Dividing both sides of the equation by cos^2a gives us tan^2a + 1 = sec^2a.
Now, we know that sec^2a = 1 + tan^2a. Substituting this into the previous equation, we get 1 + tan^2a = tan^2a sin^2a.
Simplifying further, we obtain tan^2a - 1 + cos^2a = tan^2a sin^2a, which proves the given identity.
29. To prove that tan B sin B + cos B = sec B, we can start with the identity sin^2B + cos^2B = 1. Dividing both sides of the equation by sin B cos B, we get tan B + 1 = sec B.
Subtracting 1 from both sides, we obtain tan B = sec B - 1. Rearranging the equation, we have tan B - sec B = -1.
Now, using the identity tan B = sin B / cos B and sec B = 1 / cos B, we can write the equation as sin B / cos B - 1 / cos B = -1. Combining the fractions with a common denominator, we get (sin B - 1) / cos B = -1.
Multiplying both sides by cos B, we obtain sin B - 1 = -cos B. Rearranging the equation, we have sin B + cos B = 1.
Recalling the Pythagorean identity sin^2B + cos^2B = 1, we see that the left side of the equation matches the identity. Therefore, sin B + cos B = 1, which proves the given identity.