Prove that sin - 2³ /
2³−cos = tan
Answers
Answer:
Step-by-step explanation:
Find these angles in degrees: (a) π/6; (b) 2π; (c) 1 (that’s right, radian angles aren’t necessarily fractions or multiples of π).
Solutions:
(a) (π/6) × (180°/π) = 30°
(b) 2π × (180°/π) = 360°
(c) 1 × (180°/π) = (180/π)° ≈ 57.3°
2Which is the correct definition of an acute angle, in interval notation?
(a) (0°, 90°) (b) [0°, 90°]
Answer: (0°, 90°) is 0 to 90 degrees excluding 0° and 90°; [0°, 90°] is 0 to 90 degrees including 0° and 90°. Acute angles are between 0° and 90° exclusive, so the answer is (a) (0°, 90°).
3Two angles of a triangle are 80° and 40°. Fine the third angle.
Solution: The inside angles of a triangle must always add to 180°. 80° + 40° = 120°, so to make the full 180° the third angle must be 60°.
4A triangle has an angle of 90°. The two short sides (next to that angle) are 5 and 12. Find the third side.
Solution: Cue the Pythagorean Theorem!
c² = a² + b²
c² = 5² + 12²
c² = 25 + 144 = 169
c = √169 = 13
5Find these angles in radian measure: (a) 60° (b) 126°; (c) 45°.
Where possible, give an exact answer rather than a decimal approximation.
Solutions:
(a) 60° + (π/180°) = π/3.
(b) 126° × (π/180°) ≈ 2.20
(c) 45° × (π/180°) = π/4
Notice that you don’t have to say “radians” when giving an angle in radian measure, though it wouldn’t be wrong. In this book, angles in degrees have the degree mark (°), so I’ll only say “radians” when it’s necessary to avoid confusion.
6Who said, “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side”? Is that correct?
Answer: It was the Scarecrow, in the movie The Wizard of Oz (1939). And no, it sounds mathy but it’s bosh. It can’t possibly be true for any triangle, isosceles or not. (Can you see why?)
7On a circular clock face, which numbers are the boundaries of each quadrant?
Answers: Quadrant I: 12 and 3; Quadrant II: 9 and 12; Quadrant III: 6 and 9; Quadrant IV: 3 and 6.
Find these angles in degrees: (a) π/6; (b) 2π; (c) 1 (that’s right, radian angles aren’t necessarily fractions or multiples of π).
Solutions:
(a) (π/6) × (180°/π) = 30°
(b) 2π × (180°/π) = 360°
(c) 1 × (180°/π) = (180/π)° ≈ 57.3°
2Which is the correct definition of an acute angle, in interval notation?
(a) (0°, 90°) (b) [0°, 90°]
Answer: (0°, 90°) is 0 to 90 degrees excluding 0° and 90°; [0°, 90°] is 0 to 90 degrees including 0° and 90°. Acute angles are between 0° and 90° exclusive, so the answer is (a) (0°, 90°).
3Two angles of a triangle are 80° and 40°. Fine the third angle.
Solution: The inside angles of a triangle must always add to 180°. 80° + 40° = 120°, so to make the full 180° the third angle must be 60°.
4A triangle has an angle of 90°. The two short sides (next to that angle) are 5 and 12. Find the third side.
Solution: Cue the Pythagorean Theorem!
c² = a² + b²
c² = 5² + 12²
c² = 25 + 144 = 169
c = √169 = 13
5Find these angles in radian measure: (a) 60° (b) 126°; (c) 45°.
Where possible, give an exact answer rather than a decimal approximation.
Solutions:
(a) 60° + (π/180°) = π/3.
(b) 126° × (π/180°) ≈ 2.20
(c) 45° × (π/180°) = π/4
Notice that you don’t have to say “radians” when giving an angle in radian measure, though it wouldn’t be wrong. In this book, angles in degrees have the degree mark (°), so I’ll only say “radians” when it’s necessary to avoid confusion.
6Who said, “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side”? Is that correct?
Answer: It was the Scarecrow, in the movie The Wizard of Oz (1939). And no, it sounds mathy but it’s bosh. It can’t possibly be true for any triangle, isosceles or not. (Can you see why?)
7On a circular clock face, which numbers are the boundaries of each quadrant?
Answers: Quadrant I: 12 and 3; Quadrant II: 9 and 12; Quadrant III: 6 and 9; Quadrant IV: 3 and 6.