calculate the following:
1)sin 50 degree-sin 70 degree+sin 10deg.
2)cos square 48 deg.- sin square 12 deg.
3)tan 20 deg.+tan 40 deg.+root 3 tan 20 tan 40
Plz. Solve these
1. sin50 - sin70 + sin10 =
0.7660 - 0.9397 + 0.1736 = -0.0001.
2. cos^2(48) - sin^2(12) =
0.4477 - 0.0432 = 0.4045.
3. tan20 + tan40 + sqrt(3 )*tan20*tan40
= 0.3640 + 0.8391 + 0.5290 = 1.7321.
9
2
To calculate these trigonometric expressions, we will use the trigonometric identities and the unit circle. Let's solve these step by step:
1) sin 50° - sin 70° + sin 10°:
To begin, we'll convert the given angles to radians, as trigonometric functions usually work with radian measurements. To convert degrees to radians, we multiply the degree value by π/180.
sin(50°) = sin(50π/180)
sin(70°) = sin(70π/180)
sin(10°) = sin(10π/180)
Now, we can directly substitute these values into the expression:
sin(50π/180) - sin(70π/180) + sin(10π/180)
To compute these trigonometric values, we need to use a reference angle within the unit circle. Let's use the unit circle to find these values:
- For sin(50π/180), we can use the reference angle of π/6 (30°).
- For sin(70π/180), we can use the reference angle of π/3 (60°).
- For sin(10π/180), we can use the reference angle of π/18 (10°).
Using the unit circle, we find:
sin(π/6) = 1/2,
sin(π/3) = √3/2,
sin(π/18) = (√3 - 1)/(2√2).
Now we can substitute in these values and simplify the expression:
(1/2) - (√3/2) + (√3 - 1)/(2√2)
To add these fractions, we need a common denominator, which is 2√2:
(√2 - √6 + √3 - 1)/(2√2)
This is the simplified form of the expression sin 50° - sin 70° + sin 10°.
2) cos²(48°) - sin²(12°):
In this expression, we have squared trigonometric functions. We'll use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to rewrite the expression:
cos²(48°) - sin²(12°) = 1 - sin²(12°) - sin²(12°)
Using the Pythagorean identity, we know that sin²(θ) = 1 - cos²(θ). Therefore, we can rewrite sin²(12°) as 1 - cos²(12°):
1 - cos²(12°) - (1 - cos²(12°))
Simplifying, we get:
2cos²(12°) - 1
3) tan(20°) + tan(40°) + √3tan(20°)tan(40°):
To solve this expression, we'll use the tangent addition formula:
tan(α + β) = (tan(α) + tan(β))/(1 - tan(α)tan(β))
Using this formula, we can rewrite the expression as:
tan(20° + 40°) = tan(60°)
The tangent of 60° is √3, so the final result is:
tan(20°) + tan(40°) + √3tan(20°)tan(40°) = √3