Chapter 12 Trigonometry (3)

12

Trigonometry (3) Contents 12.1 Angle between Two Straight Lines 12.2 Angle between a Straight Line and a Plane 12.3 Angle between Two Planes

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12 Trigonometry (3) 12.1 Angle between Two Straight Lines The angle between two intersecting straight lines l1 and l2 is given by the acute angle between the two straight lines as shown in Fig. 12.6. If two straight lines on a plane do not intersect, they are parallel.

Fig 12.6 Home Content

Fig 12.8

In three dimensions, when two lines do intersect, we can simply consider the plane containing these intersecting lines, as shown in Fig. 12.8. In three dimensions, the angle between two intersecting straight lines is the acute angle between the straight lines lying on the same plane,

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12 Trigonometry (3) 12.1 Angle between Two Straight Lines Example 12.1 Fig.12.10 shows a right pyramid VABCD with a square base. The length of each slant edge is 10 cm and each side of the base is 8 cm long. VN is the height of the right pyramid. (a)

Find the angle between the line VB and the line VC.

(b)

Find the angle between the line VB and the line VD.

Fig 12.10

(Give the answers correct to 3 significant figures.) Home Content

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12 Trigonometry (3) 12.1 Angle between Two Straight Lines Solution: (a)

Consider a triangle with sides VB and VC, in this case, ∆VBC.

BC 2 = VB 2 + VC 2 − 2(VB )(VC ) cos ∠BVC 82 = 10 2 + 10 2 − 2(10)(10) cos ∠BVC 64 = 100 + 100 − 200 cos ∠BVC 136 200 ∠BVC ≈ 47.1563° = 47.2° (correct to 3 significant figures)

Fig 12.10

cos ∠BVC = Home Content

Since ∆VBC is an isosceles triangle, students may use another method by drawing the altitude bisecting ∆VBC and using simple trigonometric ratio to find the required angle.

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12 Trigonometry (3) 12.1 Angle between Two Straight Lines Solution: (b)

Consider a triangle with sides VB and VD, in this case, ∆VBD. DB 2 = 82 + 82 DB = 128 DN = 4 2 DN 4 2 = DV 10 ∠DVN ≈ 34.4499°

sin ∠DVN = Home Content

Fig 12.10

∠BVD = 2∠DVN ≈ 68.8998 = 68.9° (correct to 3 significant figures)

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12 Trigonometry (3) 12.2 Angle between a Straight Line and a Plane When a line cuts a plane at a point P like a javelin hitting a lawn as shown in Fig. 12.31, the angle between the line and the plane is defined as the angle between PA and PB, where PB is the projection of PA on the plane.

Fig 12.31 Home Content

In three dimensions, the angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane. PA, PB and the vertical line AB form a right-angled triangle perpendicular to the plane. Therefore, it will be easier to find the angle between a straight line and a plane if we identify a right-angled triangle involved.

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12 Trigonometry (3) 12.3 Angle between Two Planes The angle between two planes is defined as the acute angle between the perpendiculars to the intersecting lines of the two planes.

Fig. 12.54

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As shown in Fig. 12.54, the intersection of the two planes is PQ. AP and CQ lie on Plane I. BP and DQ lie on Plane II. AP ⊥ PQ and BP ⊥ PQ. Therefore, ∠APB(θ) is the angle between two planes. In three dimensions, the angle between two planes is the acute angle between two perpendiculars on the respective planes to the intersection of the two planes.

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