10th grade Exact sine, cosine and tangent of 30°, 45° and 60°, and the reciprocal ratios worksheet

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

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

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

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

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

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

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

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

   In Jun's right triangle, cos θ = 5/13 for an acute angle θ. What is cot(90° − θ)?

   • 13/12
   • 13/5
   • 5/12
   • 12/5
5. 5.

   In Luz's right triangle, cot θ = 7/24 for an acute angle θ. What is csc(90° − θ)?

   • 24/7
   • 25/7
   • 25/24
   • 24/25
6. 6.

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

   • Disagree: the expression is a sum, 2√3/3 + 1/2, so the claim of 1 fails.
   • Disagree: sec and cos multiply to 1 only for the same angle; here (2√3/3)(1/2) = √3/3.
   • Disagree: sec 30° = √2, so sec 30° · cos 60° = √2 · 1/2 = √2/2, not 1.
   • Agree: a secant and a cosine are flips of each other, so their product is always 1.
7. 7.

   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 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.
   • Disagree: the 30° ramp rises more, 5√3 m against 5 m.
8. 8.

   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 = 1 ÷ sin θ, and sin θ < 1 for every acute angle, so csc θ is above 1.
   • Agree: because sin θ is less than 1 for acute angles, csc θ must also be less than 1.
   • Agree: csc 30° = 1/2, since sin 30° = 1/2, so some acute angle has csc θ below 1.
   • Disagree: csc θ is exactly 1 for every acute angle, because csc θ is the flip of sin θ.
9. 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 m
   • 6 m
   • 6√3 + 6 m
10. 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 + 6√3 in
    • 6 + 3√3 in
    • 6 + 3√2 in
    • 9 in