When Does Cos Equal 1/2 )= And Arccos(1/2)=, Why Are They Both Equal To Pi/3

Definition of cosineThe cosine of an angle is defined as the sine of the complementary angle. The complementary angle equals the given angle subtracted from a right angle, 90°. For instance, if the angle is 30°, then its complement is 60°. Generally, for any angle θ,cosθ=sin(90°–θ).Written in terms of radian measurement, this identity becomescosθ=sin(π/2–θ).Right triangles and cosinesConsider a right triangle ABC with a right angle at C.

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As mentioned before, we’ll generally use the letter a to denote the side opposite angle A, the letter b to denote the side opposite angle B, and the letter c to denote the side opposite angle C.Since the sum of the angles in a triangle equals 180°, and angle C is 90°, that means angles A and B add up to 90°, that is, they are complementary angles. Therefore the cosine of B equals the sine of A. We saw on the last page that sinA was the opposite side over the hypotenuse, that is, a/c. Hence, cosB equals a/c. In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:Also, cosA=sinB=b/c.The Pythagorean identity for sines and cosinesRecall the Pythagorean theorem for right triangles. It says thata2+b2=c2where c is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by c2 and you geta2/c2+b2/c2=1.But a2/c2=(sinA)2, and b2/c2=(cosA)2. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin2 A is an abbreviation for (sinA)2, and similarly for powers of the other trig functions. Thus, we have proven thatsin2 A+cos2 A=1when A is an acute angle. We haven’t yet seen what sines and cosines of other angles should be, but when we do, we’ll have for any angle θ one of most important trigonometric identities, the Pythagorean identity for sines and cosines:Sines and cosines for special common anglesWe can easily compute the sines and cosines for certain common angles. Consider first the 45° angle. It is found in an isosceles right triangle, that is, a 45°-45°-90° triangle. In any right triangle c2=a2+b2, but in this one a=b, so c2=2a2. Hence c=a√2. Therefore, both the sine and cosine of 45° equal 1/√2 which may also be written √2/2.
Next consider 30° and 60° angles. In a 30°-60°-90° right triangle, the ratios of the sides are 1:√3:2. It follows that sin30°=cos60°=1/2, and sin60°=cos30°=√3/2.These findings are recorded in this table.

	90°	π/2	0	1
	60°	π/3	1/2	√3 / 2
	45°	π/4	√2 / 2	√2 / 2
	30°	π/6	√3 / 2	1/2
	0°	0	1	0

ExercisesThese exercises all refer to right triangles with the standard labelling.30. b=2.25 meters and cosA=0.15. Find a and c.33. b=12 feet and cosB=1/3. Find c and a.35. b=6.4, c=7.8. Find A and a.36. A=23° 15″, c=12.15. Find a and b.Hints30. The cosine of A relates b to the hypotenuse c, so you can first compute c. Once you know b and c, you can find a by the Pythagorean theorem.

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33. You know b and cosB. Unfortunately, cosB is the ratio of the two sides you don’t know, namely, a/c. Still, this gives you an equation to work with: 1/3=a/c. Then c=3a. The Pythagorean theorem then implies that a2+144=9a2. You can solve this last equation for a and then find c.35. b and c give A by cosines and a by the Pythagorean theorem.36. A and c give a by sines and b by cosines.Answers30. c=b/cosA = 2.25/0.15 =15 meters; a=14.83 meters.33. 8a2=144, so a2=18. Therefore a is 4.24″, or 4″3″.c=3a which is 12.73″, or 12″9″.35. cosA=b/c=6.4/7.8=0.82. Therefore A=34.86° = 34°52″, or about 35°.a2=7.82–6.42 = 19.9, so a is about 4.5.36.

a=csinA = 12.15sin23°15″ = 4.796.b=ccosA = 12.15cos23°15″ = 11.17.