sin 45° ≈
sin 27° ≈
sin 32,8° ≈
sin 3,45° ≈
sin 68°50′ ≈
sin 5°5′ ≈
Answers
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°
2 Which 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°).
3 Two 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°.
4 A 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
5 Find 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.
6 Who 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?)
7 On 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.
Solutions for Part 2: The Six Functions
130-60-90 degree triangle. Hypotenuse is 2 units long, and short side is 1 unit long. Find all six functions of the angle 30°. Find sine, cosine, and tangent of 60°.
30-60-90 degree triangle. Hypotenuse is 2 units long, and short side is 1 unit long.Solution: First off, you need the length of the horizontal side. You remember the theorem of Pythagoras: 1² + b² = 2², from which you get b = √3. After that, it’s just a matter of remembering the definitions. If you need a refresher, you’ll find sine and cosine at equation 1, tangent at equation 4, and the others at equation 5.
sin 30° = 1/2
cos 30° = (√3)/2
tan 30° = 1/√3 or (√3)/3
cot 30° = 1/(1/√3) = √3
sec 30° = 1/(√3/2) = 2/√3 or (2 √3)/3
csc 30° = 1/(1/2) = 2
sin 60° = (√3)/2
cos 60° = 1/2
tan 60° = √3/1 = √3