Plane Trigonometry for the Use of Students Preparing for Examinations: Containing the More Advanced Propositions, Solution of Problems and a Complete Summary of Formulae, Bookwork, Etc., Together with Recent Examination Papers for the Army, Woolwich, India, and Home Civil Services, Etc. With Answers

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Rivingtons, 1885 - Plane trigonometry - 368 pages
 

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Page 296 - 2 A. 3. Find a general expression for all the angles which have the same tangent as a given angle. Solve the equation— * 4. If A + B be less than a right angle, prove, by means of geometrical figures, the formulae— sin (A + B) = sin A cos B + cos A sin B, sin A + sin B=2 sin
Page 209 - that the angle subtended at the centre of a circle by an arc equal in length to its radius is an invariable angle. One angle of a triangle is 45, and the circular measure of another is
Page 53 - Since sin 18 is +, we take the upper sign, .'. sin 18 (ii.) Geometrically. = cos 72. Let ABC be a triangle having each of the angles at the base double of the angle at the vertex.
Page 186 - XIII. 1. The angle subtended at the centre of a circle by an arc which is equal in length to the radius is invariable. Express in degrees and circular measure the vertical angle of an isosceles triangle -which is half of each of the angles at the base. * 2. Investigate an expression which shall include all angles that have a given cosine. 3. Prove the following
Page 206 - XXXII. 1. Assuming that a circle may be treated as a regular polygon with an infinite number of sides, show that the ratio of the circumference of a circle to its diameter is constant. What is the circular measure of the least angle whose sine is — , and what is the measure in degrees, etc., of 2 the angle whose circular measure is
Page 228 - 26c cos A, and apply it to prove that if the straight line which bisects the vertical angle of a triangle also bisects the base, then the triangle must be isosceles. 9. Find the area of a triangle in terms of the sides. 10. Find the radius of the circle which touches one side of a triangle
Page 297 - when expressed in terms of sin A, has four values. 7. Prove that in a triangle the sides are proportional to the sines of the opposite angles. If, in a triangle ABC,
Page 209 - and 45 respectively, and the length of the side opposite to the latter is a furlong. Show that the field contains exactly two acres and a half. 10. Find an expression for the diameter of the circle which touches one side of a triangle and the other sides produced. If
Page 272 - cos (A - B) = cos A cos B + sin A sin B, and show that, if the tangent of one angle of a triangle be equal to the sum of the sines of the other two, the tangents of the halves of the angles will be in geometrical progression. 2. Find an expression for the area of a
Page 196 - of cot 90, tan 180, tan .270. Show that the tangent of any angle will have the same sign as its cotangent. 3. Prove by means of a geometrical construction— cos (A + B) = cos A cos B - sin A sin B, and explain the result if

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