This result might have been obtained very easily from Art. 84. For, in this triangle, 88. In any triangle the following formula is true: Thus the formula will be proved, if we prove that The angles A and B may be written in the following equivalent forms: 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. 11(cosA+cosC)= 20 cosB. 11(sin2A + sin 2C) = 19 sin2B. 2. In a triangle a=15, b=26, c=37; find (Art. 83). 3. ABC is a triangle in which the angle A is equal to the 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 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 sin B sinA a sinCsinA B 38 a C b |