evaluate without using a calculator, leaving your answer as surds
(a)sin105 degrees
(b)tan15 degrees
(c)cos 75 degrees
Oops: sin 105 = cos 15
(a) To evaluate sin 105 degrees without using a calculator, we can use the following trigonometric identity:
sin(180 - θ) = sin θ
Therefore, sin 105 degrees can be written as sin (180 - 105) degrees, which simplifies to sin 75 degrees.
(b) To evaluate tan 15 degrees without using a calculator, we can use the following trigonometric identity:
tan(−θ) = −tan θ
Therefore, tan 15 degrees can be written as -tan (-15) degrees, which is equal to -tan 15 degrees.
(c) To evaluate cos 75 degrees without using a calculator, we can use the following trigonometric identity:
cos(180 - θ) = -cos θ
Therefore, cos 75 degrees can be written as -cos (180 - 75) degrees, which is equal to -cos 105 degrees.
To evaluate trigonometric functions without using a calculator, we need to rely on our knowledge of trigonometric values and identities. Let's evaluate each expression step by step:
(a) sin 105 degrees:
We can start by using the angle sum formula for sine: sin(α + β) = sin α · cos β + cos α · sin β. In this case, we can use 45 degrees and 60 degrees as our angles:
sin 105 degrees = sin (45 degrees + 60 degrees)
Now, we substitute the values into the formula:
sin 105 degrees = (sin 45 degrees · cos 60 degrees) + (cos 45 degrees · sin 60 degrees)
Using our knowledge of the unit circle and trigonometric values, we can substitute the values:
sin 105 degrees = (√2/2 · 1/2) + (√2/2 · √3/2)
Simplifying each term:
sin 105 degrees = (√2/4) + (√6/4)
Combining the terms:
sin 105 degrees = (√2 + √6)/4
So, the value of sin 105 degrees, when expressed as a surd, is (√2 + √6)/4.
(b) tan 15 degrees:
We can use the tangent half-angle formula: tan (α/2) = (1 - cos α) / sin α. In this case, we can use 30 degrees as our angle:
tan 15 degrees = tan (30 degrees / 2)
Now, we substitute the value into the formula:
tan 15 degrees = (1 - cos 30 degrees) / sin 30 degrees
Using our knowledge of the unit circle and trigonometric values, we can substitute the values:
tan 15 degrees = (1 - √3/2) / 1/2
Simplifying the expression:
tan 15 degrees = (2 - √3) / 1
So, the value of tan 15 degrees, when expressed as a surd, is (2 - √3).
(c) cos 75 degrees:
We can use the angle sum formula for cosine: cos(α + β) = cos α · cos β - sin α · sin β. In this case, we can use 30 degrees and 45 degrees as our angles:
cos 75 degrees = cos (30 degrees + 45 degrees)
Now, we substitute the values into the formula:
cos 75 degrees = (cos 30 degrees · cos 45 degrees) - (sin 30 degrees · sin 45 degrees)
Using our knowledge of the unit circle and trigonometric values, we can substitute the values:
cos 75 degrees = (√3/2 · √2/2) - (1/2 · √2/2)
Simplifying each term:
cos 75 degrees = (√6/4) - (√2/4)
Combining the terms:
cos 75 degrees = (√6 - √2)/4
So, the value of cos 75 degrees, when expressed as a surd, is (√6 - √2)/4.
sin 15 = sin 30/2
= √[(1-cos 30)/2]
= 1/2 √(2-√3)
cos 15 = 1/2 √(2+√3)
sin 105 = sin(90+15) = sin 90 cos 15 - cos 90 sin 15 = 1/2 √(2-√3)
tan 15 = √(2-√3)/√(2+√3) = 2-√3 = 1/(2+√3)
cos 75 = cos(90-15) = cos90cos15 + sin90 sin15 = 1/2 √(2-√3)