Reciprocal Identities Practice Quiz (HSF-TF.C.9)

1) Simplify the expression using reciprocal identities: sinθ · cscθ

Sin²θ

Csc²θ

1

Secθ

Step 1: Use the reciprocal identity for cscθ = 1/sinθ.

Step 2: Substitute into the expression: sinθ × (1/sinθ) = 1.

So, the final answer is 1.

Explanation

Step 1: Use the reciprocal identity for cscθ = 1/sinθ.

Step 2: Substitute into the expression: sinθ × (1/sinθ) = 1.

So, the final answer is 1.

2) Simplify: (1 + secθ)/secθ

1 − cosθ

1 + secθ

Cosθ + 1

1 + cosθ divided by cosθ

Step 1: Rewrite secθ as 1/cosθ.

Step 2: Substitute: (1 + 1/cosθ) / (1/cosθ).

Step 3: Multiply numerator and denominator by cosθ:

[(cosθ + 1)/cosθ] × cosθ = 1 + cosθ.

So, the final answer is (1 + cosθ)/cosθ.

Explanation

Step 1: Rewrite secθ as 1/cosθ.

Step 2: Substitute: (1 + 1/cosθ) / (1/cosθ).

Step 3: Multiply numerator and denominator by cosθ:

[(cosθ + 1)/cosθ] × cosθ = 1 + cosθ.

So, the final answer is (1 + cosθ)/cosθ.

3) If sinθ = −4/7 and θ in QIII, find cscθ.

−7/4

7/4

−4/7

4/7

Step 1: Use reciprocal identity cscθ = 1/sinθ.

Step 2: Substitute: 1 / (−4/7) = −7/4.

So, the final answer is −7/4.

Explanation

Step 1: Use reciprocal identity cscθ = 1/sinθ.

Step 2: Substitute: 1 / (−4/7) = −7/4.

So, the final answer is −7/4.

4) If cosθ = −3/5 and θ in QII, find secθ.

3/5

−3/5

−5/3

5/3

Step 1: secθ = 1/cosθ = 1 / (−3/5) = −5/3.

Step 2: In Quadrant II, cos is negative, so sec is also negative.

So, the final answer is −5/3.

Explanation

Step 1: secθ = 1/cosθ = 1 / (−3/5) = −5/3.

Step 2: In Quadrant II, cos is negative, so sec is also negative.

So, the final answer is −5/3.

5) Simplify: sinθ/(1/cscθ)

Sin²θ

1

Csc²θ

Secθ

Step 1: Replace 1/cscθ with sinθ.

Step 2: The expression becomes sinθ × sinθ = sin²θ.

So, the final answer is sin²θ.

Explanation

Step 1: Replace 1/cscθ with sinθ.

Step 2: The expression becomes sinθ × sinθ = sin²θ.

So, the final answer is sin²θ.

6) Right triangle: opposite = 3, hypotenuse = 6. Which equals cscθ?

1/2

2

3/5

5/3

Step 1: sinθ = opposite / hypotenuse = 3/6 = 1/2.

Step 2: cscθ = 1/sinθ = 1 / (1/2) = 2.

So, the final answer is 2.

Explanation

Step 1: sinθ = opposite / hypotenuse = 3/6 = 1/2.

Step 2: cscθ = 1/sinθ = 1 / (1/2) = 2.

So, the final answer is 2.

7) If tanθ = 7, what is cotθ?

7

1/7

−7

−1/7

Step 1: cotθ = 1/tanθ = 1/7.

So, the final answer is 1/7.

Explanation

Step 1: cotθ = 1/tanθ = 1/7.

So, the final answer is 1/7.

8) Simplify: cscθ − 1/sinθ

0

Cotθ

Tanθ

Secθ

Step 1: cscθ = 1/sinθ.

Step 2: 1/sinθ − 1/sinθ = 0.

So, the final answer is 0.

Explanation

Step 1: cscθ = 1/sinθ.

Step 2: 1/sinθ − 1/sinθ = 0.

So, the final answer is 0.

9) If cscθ = 13/5 and θ in QII, what is sinθ?

12/13

−5/13

5/13

−12/13

Step 1: sinθ = 1/cscθ = 5/13.

Step 2: Since θ is in Quadrant II, sinθ is positive.

So, the final answer is 5/13.

Explanation

Step 1: sinθ = 1/cscθ = 5/13.

Step 2: Since θ is in Quadrant II, sinθ is positive.

So, the final answer is 5/13.

10) Simplify: secθ·cosθ + cscθ·sinθ

1

2

Sin²θ + cos²θ

0

Step 1: secθ·cosθ = 1 and cscθ·sinθ = 1.

Step 2: 1 + 1 = 2.

So, the final answer is 2.

Explanation

Step 1: secθ·cosθ = 1 and cscθ·sinθ = 1.

Step 2: 1 + 1 = 2.

So, the final answer is 2.

11) Rewrite tanθ using only sine and cosine.

1/cosθ

Sinθ/cosθ

1/sinθ

Cosθ/sinθ

Step 1: tanθ = sinθ/cosθ.

So, the final answer is sinθ/cosθ.

Explanation

Step 1: tanθ = sinθ/cosθ.

So, the final answer is sinθ/cosθ.

12) Use reciprocal identities to simplify: secθ · cosθ

1

Tanθ

Sinθ

Cscθ

Step 1: secθ = 1/cosθ.

Step 2: (1/cosθ) × cosθ = 1.

So, the final answer is 1.

Explanation

Step 1: secθ = 1/cosθ.

Step 2: (1/cosθ) × cosθ = 1.

So, the final answer is 1.

13) Simplify: sinθ/cscθ + cosθ/secθ

0

1

2

Sinθ + cosθ

Step 1: Rewrite the expression using reciprocal identities.

The given expression is (sinθ / cscθ) + (cosθ / secθ).

We know cscθ = 1/sinθ and secθ = 1/cosθ.

Substitute these identities: (sinθ / (1/sinθ)) + (cosθ / (1/cosθ)).

Step 2: Simplify each term.

Multiply numerator by reciprocal of the denominator: (sinθ × sinθ) + (cosθ × cosθ) = sin²θ + cos²θ.

Step 3: Apply the Pythagorean identity.

sin²θ + cos²θ = 1.

So, the final answer is 1.

Explanation

Step 1: Rewrite the expression using reciprocal identities.

The given expression is (sinθ / cscθ) + (cosθ / secθ).

We know cscθ = 1/sinθ and secθ = 1/cosθ.

Substitute these identities: (sinθ / (1/sinθ)) + (cosθ / (1/cosθ)).

Step 2: Simplify each term.

Multiply numerator by reciprocal of the denominator: (sinθ × sinθ) + (cosθ × cosθ) = sin²θ + cos²θ.

Step 3: Apply the Pythagorean identity.

sin²θ + cos²θ = 1.

So, the final answer is 1.

14) Which simplifies to 1 for all θ where defined?

Secθ + cosθ

Tanθ·cotθ

Sinθ + cscθ

Cscθ − sinθ

Step 1: Use reciprocal identity cotθ = 1/tanθ.

Step 2: Substitute: tanθ × (1/tanθ) = 1.

Explanation

Step 1: Use reciprocal identity cotθ = 1/tanθ.

Step 2: Substitute: tanθ × (1/tanθ) = 1.

15) If csc α = 5/13 and α is acute, find sec α.

13/12

5/13

13/5

5/12

Step 1: sinα = 1/cscα = 13/5 → This is reversed. Actually, sinα = 1/(5/13) = 13/5 is incorrect. Let’s fix it.

Step 1: sinα = 1/cscα = 1 / (5/13) = 13/5 (impossible, greater than 1).

Hence cscα = 13/5 instead of 5/13.

Now sinα = 5/13.

Step 2: Use the Pythagorean identity cos²α + sin²α = 1.

cos²α = 1 − (5/13)² = 1 − 25/169 = 144/169 → cosα = 12/13.

Step 3: secα = 1/cosα = 13/12.

So, the final answer is 13/12.

Explanation

Step 1: sinα = 1/cscα = 13/5 → This is reversed. Actually, sinα = 1/(5/13) = 13/5 is incorrect. Let’s fix it.

Step 1: sinα = 1/cscα = 1 / (5/13) = 13/5 (impossible, greater than 1).

Hence cscα = 13/5 instead of 5/13.

Now sinα = 5/13.

Step 2: Use the Pythagorean identity cos²α + sin²α = 1.

cos²α = 1 − (5/13)² = 1 − 25/169 = 144/169 → cosα = 12/13.

Step 3: secα = 1/cosα = 13/12.

So, the final answer is 13/12.

16) Unit circle point P(−a, −b) in QIII (a,b>0). Which is true?

Secθ > 0 and cscθ > 0

Secθ < 0 and cscθ > 0

Secθ > 0 and cscθ < 0

Secθ < 0 and cscθ < 0

Step 1: In Quadrant III, both sine and cosine are negative.

Step 2: secθ = 1/cosθ → negative; cscθ = 1/sinθ → negative.

So, the final answer is secθ < 0 and cscθ < 0.

Explanation

Step 1: In Quadrant III, both sine and cosine are negative.

Step 2: secθ = 1/cosθ → negative; cscθ = 1/sinθ → negative.

So, the final answer is secθ < 0 and cscθ < 0.

17) Simplify: 1 + tanθ · cotθ

1

2

Sinθ

Sec²θ

Step 1: Use cotθ = 1/tanθ.

Step 2: Substitute: 1 + tanθ × (1/tanθ) = 1 + 1 = 2.

So, the final answer is 2.

Explanation

Step 1: Use cotθ = 1/tanθ.

Step 2: Substitute: 1 + tanθ × (1/tanθ) = 1 + 1 = 2.

So, the final answer is 2.

18) Solve tan x = √3, 0 ≤ x < 2π.

X = π/3, 4π/3

X = π/6, 7π/6

X = π/3, 2π/3

X = π/4, 5π/4

Step 1: tanx = √3 means reference angle = π/3.

Step 2: Tangent is positive in Quadrants I and III.

So, x = π/3 and 4π/3.

So, the final answer is x = π/3 and 4π/3.

Explanation

Step 1: tanx = √3 means reference angle = π/3.

Step 2: Tangent is positive in Quadrants I and III.

So, x = π/3 and 4π/3.

So, the final answer is x = π/3 and 4π/3.

19) Simplify (secθ + tanθ)(secθ − tanθ)

Sec²θ − tan²θ

Sin²θ

Cos²θ

1

Step 1: Use difference of squares formula.

(secθ + tanθ)(secθ − tanθ) = sec²θ − tan²θ.

Step 2: Use the identity sec²θ − tan²θ = 1.

So, the final answer is 1.

Explanation

Step 1: Use difference of squares formula.

(secθ + tanθ)(secθ − tanθ) = sec²θ − tan²θ.

Step 2: Use the identity sec²θ − tan²θ = 1.

So, the final answer is 1.

20) A ladder leans against a wall making an angle θ with the ground. The angle satisfies tanθ = 3/4. Find cotθ and secθ.

Cotθ = 3/4, secθ = 5/4

Cotθ = 4/3, secθ = 5/4

Cotθ = 4/3, secθ = 3/5

Cotθ = 3/4, secθ = 4/5

Step 1: tanθ = 3/4 ⇒ opposite = 3, adjacent = 4, hypotenuse = 5.

Step 2: cotθ = adjacent / opposite = 4/3 and secθ = hypotenuse / adjacent = 5/4.

So, the final answer is cotθ = 4/3 and secθ = 5/4.

Explanation

Step 1: tanθ = 3/4 ⇒ opposite = 3, adjacent = 4, hypotenuse = 5.

Step 2: cotθ = adjacent / opposite = 4/3 and secθ = hypotenuse / adjacent = 5/4.

So, the final answer is cotθ = 4/3 and secθ = 5/4.