Worksheet

Exact sine, cosine and tangent of 30°, 45° and 60°, and the reciprocal ratios

Grade 10 Math Worksheet - Challenge Me

NameDate

Circle the best answer.

  1. Ari works out sec²45° + tan²60° exactly. What is the value?

    • 7/2
    • 5
    • √2 + √3
    • 3

  2. Bo works out csc²60° + cot²30° exactly. What is the value?

    • 5√3/3
    • 7/3
    • 13/12
    • 13/3

  3. Esme works out sin²60° + sec²60° exactly. What is the value?

    • 25/12
    • (√3 + 4)/2
    • 19/4
    • 1

  4. In Kira's right triangle, sin θ = 8/17 for an acute angle θ. What is sec(90° − θ)?

    • 17/15
    • 17/8
    • 15/8
    • 15/17

  5. Milo claims sec 30° · cos 60° = 1, because secant is the flip of cosine. Which verdict is right?

    • Disagree: sec and cos multiply to 1 only for the same angle; here (2√3/3)(1/2) = √3/3.
    • Agree: a secant and a cosine are flips of each other, so their product is always 1.
    • Disagree: the expression is a sum, 2√3/3 + 1/2, so the claim of 1 fails.
    • Disagree: sec 30° = √2, so sec 30° · cos 60° = √2 · 1/2 = √2/2, not 1.

  6. Noah has two 10-m ramps, one at 30° and one at 60° to the ground. Noah says the 60° ramp rises exactly twice as high. Which verdict is right?

    • Disagree: the 30° ramp rises more, 5√3 m against 5 m.
    • Disagree: the ramps rise 10 tan 30° and 10 tan 60°, about 5.8 m and 17.3 m.
    • Disagree: heights are 10 sin 30° = 5 m and 10 sin 60° = 5√3 m, about 8.7 m.
    • Agree: 60° is twice 30°, so the height doubles from 5 m to 10 m.

  7. Pia claims some acute angle θ has csc θ less than 1, because sin θ is less than 1. Which verdict is right?

    • Disagree: csc θ is exactly 1 for every acute angle, because csc θ is the flip of sin θ.
    • Disagree: csc θ = 1 ÷ sin θ, and sin θ < 1 for every acute angle, so csc θ is above 1.
    • Agree: csc 30° = 1/2, since sin 30° = 1/2, so some acute angle has csc θ below 1.
    • Agree: because sin θ is less than 1 for acute angles, csc θ must also be less than 1.

  8. Ren sees a tower's top at 45° from point A on level ground. After walking 20 m straight back to B, the angle is 30°. How tall is the tower? Ignore Ren's height.

    • 20 m
    • 20√3 m
    • 10 + 10√3 m
    • 10√3 − 10 m

  9. Tala's 12-m ladder leans on a wall at 60° to the ground. The foot is then pulled out until the angle is 30°. How far does the foot slide?

    • 6√3 + 6 m
    • 6√3 − 6 m
    • 6√3 m
    • 6 m

  10. Zee draws a house sign: a square, 6 in on each side, with an equilateral triangle roof on top, sharing the top side. How tall is the sign, from bottom to peak?

    • 6 + 3√2 in
    • 6 + 3√3 in
    • 9 in
    • 6 + 6√3 in
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Exact sine, cosine and tangent of 30°, 45° and 60°, and the reciprocal ratios

Answer Key - Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

  1. Ari works out sec²45° + tan²60° exactly. What is the value?

    • 7/2
    • 5
    • √2 + √3
    • 3

  2. Bo works out csc²60° + cot²30° exactly. What is the value?

    • 5√3/3
    • 7/3
    • 13/12
    • 13/3

  3. Esme works out sin²60° + sec²60° exactly. What is the value?

    • 25/12
    • (√3 + 4)/2
    • 19/4
    • 1

  4. In Kira's right triangle, sin θ = 8/17 for an acute angle θ. What is sec(90° − θ)?

    • 17/15
    • 17/8
    • 15/8
    • 15/17

  5. Milo claims sec 30° · cos 60° = 1, because secant is the flip of cosine. Which verdict is right?

    • Disagree: sec and cos multiply to 1 only for the same angle; here (2√3/3)(1/2) = √3/3.
    • Agree: a secant and a cosine are flips of each other, so their product is always 1.
    • Disagree: the expression is a sum, 2√3/3 + 1/2, so the claim of 1 fails.
    • Disagree: sec 30° = √2, so sec 30° · cos 60° = √2 · 1/2 = √2/2, not 1.

  6. Noah has two 10-m ramps, one at 30° and one at 60° to the ground. Noah says the 60° ramp rises exactly twice as high. Which verdict is right?

    • Disagree: the 30° ramp rises more, 5√3 m against 5 m.
    • Disagree: the ramps rise 10 tan 30° and 10 tan 60°, about 5.8 m and 17.3 m.
    • Disagree: heights are 10 sin 30° = 5 m and 10 sin 60° = 5√3 m, about 8.7 m.
    • Agree: 60° is twice 30°, so the height doubles from 5 m to 10 m.

  7. Pia claims some acute angle θ has csc θ less than 1, because sin θ is less than 1. Which verdict is right?

    • Disagree: csc θ is exactly 1 for every acute angle, because csc θ is the flip of sin θ.
    • Disagree: csc θ = 1 ÷ sin θ, and sin θ < 1 for every acute angle, so csc θ is above 1.
    • Agree: csc 30° = 1/2, since sin 30° = 1/2, so some acute angle has csc θ below 1.
    • Agree: because sin θ is less than 1 for acute angles, csc θ must also be less than 1.

  8. Ren sees a tower's top at 45° from point A on level ground. After walking 20 m straight back to B, the angle is 30°. How tall is the tower? Ignore Ren's height.

    • 20 m
    • 20√3 m
    • 10 + 10√3 m
    • 10√3 − 10 m

  9. Tala's 12-m ladder leans on a wall at 60° to the ground. The foot is then pulled out until the angle is 30°. How far does the foot slide?

    • 6√3 + 6 m
    • 6√3 − 6 m
    • 6√3 m
    • 6 m

  10. Zee draws a house sign: a square, 6 in on each side, with an equilateral triangle roof on top, sharing the top side. How tall is the sign, from bottom to peak?

    • 6 + 3√2 in
    • 6 + 3√3 in
    • 9 in
    • 6 + 6√3 in