A. Incorrect! This equality is true for all values of x. Therefore, this is an identity and not a conditional equation.

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Question No. 1 of 10

Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.

Question

1. Which of the following equalities is a conditional equation?

(A) ⁵

5 x = x

(B) 3x – x = 2x (C) 5 1 4

x + = −

(D) cos(-x) = cos x (E) cotx = tan¹x

Feedback

A. Incorrect!

This equality is true for all values of x. Therefore, this is an identity and not a conditional equation.

B. Incorrect!

C. Correct!

The only number that makes this equality true is -25. Therefore, this equality is a conditional equation and not an identity.

D. Incorrect!

E. Incorrect!

Solution

An identity is true for all defined values of x, whereas only certain numbers satisfy a conditional equation. The only number that makes 5 1 4

x + = − true is -25.

Therefore, it is a conditional equation and not an identity.

(C) 1 4 5

x + = −

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needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.

Question

2. Which of the following equalities is an identity?

(A) cos² x + sin² x = 1 (B) sec x = 2

(C) 4x + 2 = 6x (D) x⁴ = 16

(E) x² + 2x – 3 = 0

Feedback

A. Correct!

Every number belonging to the domains of cosine and sine satisfy this equality.

Therefore, this equality is an identity.

B. Incorrect!

This equality has an infinite number of solutions, but not every number belonging to the domain of secant will satisfy this equality.

C. Incorrect!

The only number that satisfies this equality is the number 1. Therefore, this equality is a conditional equation and not an identity.

D. Incorrect!

The only numbers that satisfy this equality are the -2 and 2. Therefore, this equality is a conditional equation and not an identity.

E. Incorrect!

The only numbers that satisfy this equality are the -3 and 1. Therefore, this equality is a conditional equation and not an identity.

Solution

An identity is true for all defined values of x, whereas only certain numbers satisfy a conditional equation. Every number belonging to the domains of cosine and sine satisfy cos² x + sin² x = 1. Therefore, this equality is an identity.

(A) cos² x + sin² x = 1

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Question

3. Which of the following is a Pythagorean identity?

(A) cscx = sin¹x

(B) ^cos

( )

2π− x =^sinx

(C) cos² x + sin² x = 1 (D) tan(-x) = -tan x (E) cot ^cossin

x = xx

Feedback

A. Incorrect!

This is one of the reciprocal identities.

B. Incorrect!

This is one of the complementary identities.

C. Correct!

This is one of the Pythagorean identities.

D. Incorrect!

This is one of the even/odd identities.

E. Incorrect!

This is one of the quotient identities.

Solution

The three Pythagorean identities are:

cos² x + sin² x = 1 1 + tan² x = sec² x 1 + cot² x = csc² x

(C) cos² x + sin² x = 1

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Question

4. Which of the following is a reciprocal identity?

(A) secx = cos¹x

(B) ^sec

( )

2π−x =^cscx

(C) 1 + cot² x = csc² x (D) cos(-x) = cos x (E) tan cos^sin

x = xx

Feedback

A. Correct!

This is one of the reciprocal identities.

B. Incorrect!

This is one of the complementary identities.

C. Incorrect!

This is one of the Pythagorean identities.

D. Incorrect!

This is one of the even/odd identities.

E. Incorrect!

This is one of the quotient identities.

Solution

The three reciprocal identities are:

cscx = sin1x

secx = cos1x

cotx = tan1x

(A) sec x = 1 / cos x

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as needed (3) Pick the answer (4) Go back to review the core concept tutorial as needed.

Question

5. Given cos 1.39 = 0.18, find sin 1.39.

(A) 0.18 (B) 0.58 (C) 0.68 (D) 0.78 (E) 0.98

Feedback

A. Incorrect!

Find sin 1.39 by using the Pythagorean identity cos² x + sin² x = 1.

B. Incorrect!

C. Incorrect!

D. Incorrect!

E. Correct!

You found sin 1.39 by using the Pythagorean identity cos² x + sin² x = 1.

Solution

cos² 1.39 + sin² 1.39 = 1 0.18² + sin² 1.39 = 1 0.0324 + sin² 1.39 = 1 sin² 1.39 = 1 - 0.0324 sin² 1.39 = 0.9676 sin 1.39 = ±√0.9676 sin 1.39 ≈ ±0.98

The angle 1.39 radians lies in the first quadrant because 0 < 1.39 < π/2 (0 < 1.39

< 1.57). Therefore, keep only the positive sign because the trigonometric function sine is positive within the first quadrant.

sin 1.39 = 0.98 (E)0.98

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Question

6. sin 0.18 = 0.18. Evaluate tan 0.18.

(A) 0.08 (B) 0.18 (C) 0.28 (D) 0.38 (E) 0.48

Feedback

A. Incorrect!

Find tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using the related quotient identity.

B. Correct!

You found tan 0.18 by first using a Pythagorean identity to find cos 0.18 and then using the related quotient identity, tan cos^sin

x = xx. C. Incorrect!

D. Incorrect!

E. Incorrect!

Solution

First calculate cos 0.18 and then calculate tan 0.18 by making use of the quotient identitytan cos^sin

x = xx.

The value of cos 0.18 may be obtained by making use of the Pythagorean identity cos² 0.18 + sin² 0.18 = 1

cos² 0.18 + 0.18² = 1 cos² 0.18 + 0.0324 = 1 cos² 0.18 = 1 - 0.0324 cos² 0.18 = 0.9676 cos 0.18 = ±√0.9676 cos 0.18 ≈ ±0.98

The angle 0.18 radians lies within the first quadrant because 0 < 0.18 < π/2 (0 <

0.18 < 1.57). Therefore, keep only the positive solution because the trigonometric function cosine is positive within the first quadrant.

cos 0.18 = 0.98

Now use a quotient identity.

sin0.18 tan 0.18 cos 0.18

0.18 0.98 0.18

=

≈

(B) 0.18

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Question

7. csc x · tan x is equal to which of the following expressions?

(A) cos x (B) cos² x (C) sec x (D) sin² x (E) cot x

Feedback

A. Incorrect!

The given expression can be rewritten using a sequence of reciprocal and quotient identities.

B. Incorrect!

C. Correct!

The given expression can be rewritten as sec x by using a sequence of reciprocal and quotient identities.

D. Incorrect!

E. Incorrect!

Solution

One of the reciprocal identities states:

cscx = sin1x

tan cossin x = xx

Rewrite and simplify the original expression using the reciprocal identity

secx = cos1x:

csc tan 1 x x sin

⋅ = x ⋅cossin x 1

cos sec

x

x x

=

(C) sec x

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Question

8. 2

1 1

sin x − is equal to which of the following expressions?

(A) cot x (B) cot² x (C) tan x (D) tan² x (E) cos² x

Feedback

A. Incorrect!

The original expression can be rewritten using a combination of a reciprocal identity and a Pythagorean identity.

B. Correct!

You rewrote the original expression using cscx = sin¹x and 1 + cot² x = csc² x.

C. Incorrect!

D. Incorrect!

E. Incorrect!

Solution

cscx = sin1x

Therefore, ² 2

csc x = sin1x

The original expression can be written as:

2 2

1 1 csc 1

sin x − = x −

One of the Pythagorean identities states:

1 + cot² x = csc² x → cot² x = csc² x – 1 Therefore, the original expression can be rewritten:

2 2

2

1 1 csc 1

sin

cot x x

x

− = −

=

(B) cot² x

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Question

9. The expression ^tan

sec csc x

x+ x can be written as an expression containing only cos x

and sin x by using a combination of:

(A) Pythagorean and reciprocal identities.

(B) Pythagorean and quotient identities.

(C) Quotient and complementary identities.

(D) Quotient and reciprocal identities.

(E) Complementary and reciprocal identities.

Feedback

A. Incorrect!

Pythagorean identities are most useful if an expression contains the square or other even power of a trigonometric function.

B. Incorrect!

Pythagorean identities are most useful if an expression contains the square or other even power, of a trigonometric function.

C. Incorrect!

Complementary identities are most useful in the evaluation of trigonometric expressions containing complementary angles.

D. Correct!

Use the quotient and reciprocal identities to rewrite the expression containing only

cos x andsin x.

E. Incorrect!

Complementary identities are most useful in the evaluation of trigonometric expressions containing complementary angles.

Solution

The original expression can be rewritten using the quotient identity tan ^sin cos x x

= x and two of the reciprocal identities: secx = cos¹x andcscx = sin¹x .

tan cossin

sec csc sec csc sin 1cos 1 cos sin

x xx

x x x x

x x

+ = +

= +

The original expression has been rewritten using only sin x and cos x. Thus, we have accomplished our goal making use of quotient and reciprocal identities.

(D)Quotient and reciprocal identities.

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Question

10. Let f(x) = sin x · cos x. Which of the following statements is true?

(A) f(x) is even for all values of x.

(B) f(x) is odd for all values of x.

(C) f(x) is neither even nor odd for all values of x.

(D) f(x) is even for positive values of x, but odd for negative values of x.

(E) f(x) is odd for positive values of x, but even for negative values of x.

Feedback

A. Incorrect!

Review the definitions for even and odd functions then try again.

B. Correct!

f(x) is an odd function.

C. Incorrect!

Review the definitions for even and odd functions then try again.

D. Incorrect!

This function is even, odd, or neither, not a combination of even or odd.

E. Incorrect!

This function is even, odd, or neither, not a combination of even or odd.

Solution

Determine whether a function is even, odd, or neither by replacing its argument (x) with the negative of its argument (-x).

f(-x) = sin(-x) · cos(-x)

The even/odd identities state that sin(-x) = -sin x and cos(-x) = cos x. Therefore,

( ) sin( ) cos( ) sin cos (sin cos )

( )

f x x x

x x

f x

− = − ⋅ −

= − ⋅

= −

The function f(x) is odd for all values of x.

(B) f(x) is odd for all values of x.