Scientific Data
H 2 O, of water than the equivalent statement using ‘mass’:
2.4 Angular Measurements
2.4.2 Degrees and radians
There are 360◦ (degrees) in a full circle.
Example 2.26
Why are there ‘360’ degrees in a circle?
The choice of ‘360’ was made when ‘fractions’ were used in calculations far more frequently than they are now. The number ‘360’ was particularly good because it can be divided by many different factors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180!
The radian is an alternative measure that is often used in calculations involving rotations (Figure 2.1).
circle radius, r
r s
q
Figure 2.1 Angle in radians.
The angle,θ , in radians is defined as the arc length, s, divided by the radius, r, of the arc.
The angle in radians is given by the simple ratio:
θ =s
r [2.10]
s= r × θ [2.11]
In a complete circle, the arc length,s, will equal the circumference of the circle= 2πr.
Hence, the angle (360◦) of a complete circle=2π r
r radians= 2π radians
360◦ = 2π radians 180◦ = π radians 90◦= π/2 radians
1 radian= 180
π degrees= 57.3 . . . degrees [2.12]
2.4.3
Conversion between degrees and radians
x in radians becomes x× 180/π in degrees [2.13]
θ in degrees becomes θ× π/180 in radians [2.14]
In Excel, to convert an angle:
• from radians to degrees, use the function DEGREES; and • from degrees to radians, use the function RADIANS.
Q2.24
Convert the following angles from degrees to radians or vice versa:
(i) 360◦ into radians (iv) 1.0 radian into degrees
(ii) 90◦ into radians (v) 2.1 radians into degrees
(iii) 170◦ into radians (vi) 3.5π radians into degrees
Example 2.27
The towns of Nairobi and Singapore both lie approximately on the equator of the Earth at longitudes 36.9◦E and 103.8◦E respectively. The radius of the Earth at the equator is 6.40× 103 km.
Calculate the distance between Nairobi and Singapore along the surface of the Earth. The equator of the Earth is the circumference of a circle with radiusr= 6.40 × 103km. Nairobi and Singapore are points on the circumference of this circle separated by an angle:
Converting this angle to radians:
θ= 66.9 × π/180 radians = 1.168 radians
The distance on the ground between Nairobi and Singapore will be given by the arc length,s, between them. Using [2.11]:
s= r × θ = 6.40 × 103× 1.168 km = 7.47 × 103 km
Q2.25
In Figure 2.2, calculate the distance that the mass rises when the drum rotates by 40◦. The radius of the drum is 10 cm.
Figure 2.2
2.4.4
Trigonometric functions
In a right-angled triangle, the longest side is the hypotenuse, H .
In the triangle shown in Figure 2.3, the angle,θ , is on the left side as shown.
The side opposite the angle is called the opposite side, O .
The side next to the angle (but not the hypotenuse) is called the adjacent side, A.
The three main trigonometric functions, sine, cosine and tangent, can be calculated by taking the ratios of sides as in equation [2.15]. Many students use a simple mnemonic to remember the correct ratios: SOHCAHTOA!
sinθ = O H [2.15] cosθ = A H tanθ =O A Example 2.28
A car travels 100 m downhill along a road that is inclined at 15◦ to the horizontal. Calculate the vertical distance through which the car travels.
The 100 m travelled by the car is the hypotenuse, H , of a right-angled triangle. The
vertical distance to be calculated is the opposite side,O, using the angle of θ= 15◦: sin(15◦)= O H = O 100 giving: O= 100 × sin(15◦)= 100 × 0.259 = 25.9 m Q2.26
A tree casts a shadow that is 15 m long when the Sun is at an angle of 30◦ above the horizon.
Calculate the height of the tree.
2.4.5
Pythagoras’s equation
H2 = O2+ A2 [2.16]
(The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.)
Q2.27
One side of a rectangular field is 100 m long, and the diagonal distance from one corner to the opposite corner is 180 m. Calculate the length of the other side of the field.
2.4.6
Small angles
When the angleθ is small (i.e. less than about 10◦ or less than about 0.2 radians) it is possible to make some approximations.
In Figure 2.4: (a) r A H O r s θ θ (b) Figure 2.4 Small angles.
• The length of the arc, s, in (a) will be approximately equal to the length of the opposite side,O, in the right-angled triangle in (b): O ≈ s.
• The lengths of the adjacent side, A, and the hypotenuse, H , in (b) will be approximately equal to the radius,r, in (a): H ≈ r and A ≈ r.
If the angleθ is measured in radians and the angle is small , then:
sin(θ )= O/H ≈ s/r = θ hence sin(θ )≈ θ
tan(θ )= O/A ≈ s/r = θ hence tan(θ )≈ θ [2.17]
cos(θ )= A/H ≈ r/r = 1 hence cos(θ )≈ 1.0
Q2.28
In the following table, use a calculator to calculate values for sin(θ ), cos(θ ) and
tan(θ ) for each of the angles listed.
Use [2.14] to calculate the angleθ in radians.
Check whether the values of sin(θ ), cos(θ ) and tan(θ ) and θ in radians agree with
The calculations forθ = 20◦ have already been performed:
θ (degrees) sin(θ ) cos(θ ) tan(θ ) θ (radians)
20 0.3420 0.9397 0.3640 0.3491 10 5 1 0 Example 2.29
A right-angled triangle has an angleθ= 5◦ and an hypotenuse of length 2.0. (i) Calculate the length of the opposite side using a trigonometric function.
We know thatθ = 5◦ andH= 2.0. Using O = H × sin(θ):
O= 2.0 × sin(θ) = 2.0 × sin(5◦)= 2.0 × 0.08716 = 0.174
(ii) Assume that the triangle is approximately the same as a thin segment of a circle with a radius equal to the hypotenuse, and estimate the length of the arc using a ‘radian’ calculation.
Convertingθ = 5◦ into radians: 5◦= 5 × π/180 radians = 0.08727 radians The arc length of a circle segment with radiusr= 2.0 is given by s = r × θ:
s= 2.0 × 0.08727 = 0.175
The calculations for a triangle with a very small angle can often be made more easily using radians than using a trigonometric function.
Q2.29
Estimate the diameter of the Moon using the following information:
The Moon is known to be 384000 km away from the Earth, and the apparent disc of the Moon subtends an angle of about 0.57◦ for an observer on the Earth – as illustrated in Figure 2.5.
Figure 2.5
Do not use a calculator, but assume that 1 radian is about 57◦ (hint: 0.57◦ is a small angle).
2.4.7
Inverse trigonometric functions
The angle can be calculated from the ratios of sides by using the ‘inverse’ functions:
θ = sin−1(O/H )
θ = cos−1(A/H ) [2.18]
θ = tan−1(O/A)
Note that the above are not the reciprocals of the various functions, e.g. sin−1(O/H ) does not
equal 1/[sin(O/H )].
The ‘inverse’ function can also be written with the ‘arc’ prefix:
θ = arcsin(O/H ) θ = arccos(A/H ) θ = arctan(O/A)
Q2.30
The three sides of a right-angle triangle have lengths, 3, 4 and 5, respectively. Calculate the value of the smallest angle in the triangle using:
(i) the sine function (ii) the cosine function (iii) the tangent function
The calculation of basic angular measurements can be carried out on a calculator. Note that it is necessary to set up the ‘mode’ of the calculator to define whether it is using degrees (DEG) or radians (RAD).
Example 2.30 gives some examples of angle calculations on a calculator.
Example 2.30
Converting 36◦ to radians using 36× π/180: 36◦ = 0.6283 . . . radians
Converting 1.3 radians to degrees using 1.3× 180/π:
1.3 radians= 74.48 . . .◦
Setting the calculator to DEG mode:
sin(1.4)= 0.024 . . . cos−1(0.21)= 77.88 . . .◦
Setting the calculator to RAD mode:
sin(1.4)= 0.986 . . . cos−1(0.21)= 1.359 . . . radians
2.4.9
Using Excel for angular measurements
When using Excel for angle calculations (see Appendix I), it is important to note that Excel
uses radians as its unit of angle, not degrees. To convert an angle,θ , in radians to degrees, use
the function DEGREES, and to convert from degrees to radians, use the function RADIANS. Alternatively it is possible to use formulae derived from [2.13] and [2.14].
Excel uses the functions SIN, COS and TAN to calculate the basic trigonometric ratios. The inverse trigonometric functions are ASIN, ACOS and ATAN.
The value ofπ in Excel is obtained by entering the expression ‘= PI()’.
Example 2.31
For the following functions and formulae in Excel:
‘= DEGREES(B4)’ converts the angle held in cell B4 from a value given in radians to a value given in degrees.
‘= B4*180/PI()’ also converts the angle held in cell B4 from a value given in radians to a value given in degrees.
‘= SIN(RADIANS(B4))’ gives the sine of the angle (in degrees) held in cell B4. ‘= DEGREES(ACOS(C3/D3))’ gives the angle (in degrees) for a triangle where the
length of the adjacent side is held in C3 and the length of the hypotenuse is held in D3.
Q2.31
Refer to Q2.26, where a tree casts a shadow that is 15 m long when the Sun is at an angle of 30◦ above the horizon.
Write out the formula that would be used in Excel to calculate the height of the tree, assuming that the length of the shadow was entered into cell D1 and the angle of the Sun in degrees was entered into cell D3.