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Example 5. Using a graphical method, solve the simultaneous equations y = 2x + 3 and a: + y = 9.
Solution. In order to graph these two equations, we need to know the gradient and the y-intercept of each one. Compare y = 2x + 3 with y = mx + c; the gradient is m = 2/1 and the y-intercept is c = 3.
For x + y = 9 we need to express it in the form y = mx + c first:
y = —x + 9 Subtracting x from both sides of the equation
m = —1/1 and c = 9 Matching this equation with y = mx + c
This process is explained more fully in the entry Gradient-Intercept Form. The graphs are drawn on the same axes, as shown in the figure. It can be seen from the figure that the two straight lines intersect at the point (2, 7). The solutions to the simultaneous equations are x = 2 and y = 7.
References: Gradient-Intercept Form, Like Terms, Linear Equation, Substitution, Variable.
References: Circular Functions, Trigonometric Graphs.
The sine rule is a set of trigonometric formulas connecting the lengths of the three sides of a triangle with the sines of its angles. It is mainly used in triangles that are not right-angled, because simpler methods are used for right-angled triangles, as explained under the entry Trigonometry. The notation used for the sine rule, and also for the cosine rule, is that the sizes of the three angles of the triangle are referred to using capital letters, say A, B, and C, and the lengths of the sides are small letters, a, b, and c (see figure a). You need to be familiar with this notation in order to use the sine rule and the cosine rule.
Notation. The length of the side opposite to angle A is called a, and similarly the side of length b is opposite to angle B and the side of length c is opposite to angle C. The sine rule states that in any triangle
a b c
sin A sin B sin C
This rule is unusual in that it contains two equal signs. In fact the sine rule represents three rules, which are
a b a c be
sin A sinJ5’ sin A sinC’ sinJ5 sinC
When we use the sine rule only one of these three rules is used at a time. The sine rule is used in a triangle for calculating the length of a side if two angles and a side are known, or for calculating the size of an angle if two sides and an angle are
416 SINE RULE
known. This may sound complicated, but in practice the procedure is straightforward, as explained in the first example, in which the length of a side is calculated.
Example 1. The line QR in figure b represents a grassy bank; a ladder QP which is 4.1 meters long has one end on the bank and its other end resting against a brick wall. The ladder makes an angle of 44° with the wall and an angle of 36° with the grassy bank. Find the distance the ladder reaches up the wall, which is the length PR. There is a peg at the point Q to stop the ladder slipping down the grassy bank.
Solution. Draw the triangle PQR, but rename it ABC, because the sine rule refers to a triangle ABC. Write down what we know about triangle ABC: A = 44°, B = 36°, and c = 4.1. In order to use the sine rule we need to pair off sides with the angles opposite to them, so we need to find the size of angle C:
C = 100° Sum of angles of triangle = 180°
The sine rule can now be used. Write
sin 36° sin 100°
b = sin 36° x
Substituting c = 4.1, C = 100°, B = 36° b c
sin 100° b = 2.4 (to 1 dp)
The ladder reaches 2.4 meters up the wall
sin B sin C Multiplying both sides of the equation by sin 36° Using a calculator
The sine rule can be used to find an angle in a triangle, as explained in the following example.
Example 2. John and Bill are using two ropes tied to the top of a heavy pole to hold it upright. John’s rope is 12 meters long and makes an angle of 35° with the level ground (see figure c). John and Bill are on eactly opposite sides of the pole. If Bill’s rope is 10 meters long, calculate the angle it makes with the ground.
SLANT HEIGHT OF CONE 417
In the triangle PJB, b = 12, j = 10, and J = 35°. Write
Substituting b = 12, j = 10, and J = 35°
sin B sin J
Inverting both sides of the equation makes the next step easier
Multiplying both sides of the equation by 12
Using inv sin on the calculator Bilks rope makes an angle of 43.5° with the level ground.
References: Cosine Rule, Trigonometry.
SLANT HEIGHT OF CONE
12 _ 10
sin 15 sin 35°
sin 15 sin 35°
12 “ 10
sin!5 = 0.68829...
B = sin“1 (0.68829...) B = 43.5° (to 1 dp)
A solid is a three-dimensional object. Some solids are made up of flat surfaces called planes, such as the cube, the cuboid, the pyramid, and the tetrahedron. These are called polyhedrons, and have faces, vertices, and edges that are related by Euler’s formula. Some solids are made up of a mixture of flat surfaces and curved surfaces, such as the hemisphere, the cone, and the cylinder. Other solids are made up of only curved surfaces, such as the sphere, the torus, and the ellipsoid. For more details about some of these solids see under their respective entries.