Imágenes de páginas
PDF
EPUB
[blocks in formation]

Ex. 2. If in the triangle ABC a = 7, b=8, c = 9, show that

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[blocks in formation]

This result might have been obtained very easily from Art. 84. For, in this triangle,

[blocks in formation]

88. In any triangle the following formula is true:

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Thus the formula will be proved, if we prove that

[blocks in formation]

The angles A and B may be written in the following equivalent forms:

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Two other formulæ, similar to this one, may be obtained by

changing the letters as in the preceding Article.

The student will have no difficulty in recalling this formula when required, if he notices that it is a symmetrical formula between the angles A and B and the sides a and b opposite these angles; and a similar remark applies to the two similar formulæ.

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

11(cosA+cosC)= 20 cosB.

11(sin2A + sin 2C) = 19 sin2B.

2. In a triangle a=15, b=26, c=37; find

(Art. 83).

[blocks in formation]

3. ABC is a triangle in which the angle A is equal to the

[blocks in formation]

4. The sides of a triangle are 13, 14, 15. Find the sines of

the angles.

5. If a=11, b=60, c=61, show that the angle C is a right angle.

6. In a triangle tan, tan3=3; find tanC.

2

7. AD is the perpendicular from the angular point A to the side BC of a triangle. Show that

2

AD=b sinC=c sinB = √s(s-a) (s—b) (s—c).

α

8. The sides of a triangle are 13 feet, 14 feet, 15 feet. Find --to the nearest inch-the perpendiculars from the angles to the opposite sides.

9. In a triangle a=8, b=5, C= 60°; find c.

10. AOG is an isosceles triangle right angled at O. The hypotenuse AG is divided into six equal parts AB, BC, CD, DE, EF, FG. If a denote either of the equal sides AO, OG.

then

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

11. BE and CF are the perpendiculars from the points B and C to the opposite sides AC and AB of a triangle ABC. Prove that

AF=b cosA,

AE=c cosA.

EF= a cosA.

Hence prove that

12. If a=10, b=17, c=21, show that the angle A is less than 30°.

SECTION II. SOLUTION OF OBLIQUE-ANGLED

TRIANGLES. APPLICATIONS.

89. Solution of Oblique-angled Triangles. There are 4 cases to consider. I. Given two angles and a side.

Let the given angles be B and C and the given side a; then A = 180° - (B+C), which determines the third angle A.

By Article 84,

b

A

[blocks in formation]

sin B sinA

a

sinCsinA

B

38

a

C

b

« AnteriorContinuar »