Functions of complementary angles

In trigonometry it is occasionally convenient to speak of the cosine, the cotangent and the cosecant as the complemen­tary functions or co-functions of the sine, the tangent and the secant respectively. Conversely, the sine, the tangent and the secant are called the co-functions of the cosine, the co­tangent and the cosecant. Recalling that two angles are said to be complementary¹ if their sum is 90°, we shall prove the theorem that a trigonometric function of an angle is equal in value to the co-function of its complementary angle.

If two positive acute angles are known to have² a trigo­nometric function of one angle equal to the trigonometric co-function of the other, then the angles are complementary.

Notes:

¹ angles are said to be complementary —кути вважаються доповнювальними

² if two positive acute angles are known to have — якщо відомо, що у двох позитивних гострих кутів

EXERCISES

I. Read the following words paying attention to the pronunciation:

tangent, cotangent, agent, join, prove, root, function, conjunction, convenient, conversely, respectively, acute, cube, curved, circle.

II. Make up sentences of your own using the words and expressions given below:

angles are known to have, angles are said to be comple­mentary, to speak of, to prove the theorem, function is equal to.

III. Answer the following questions:

1. What do we call the co-functions of the cosine, the co­tangent and the cosecant? 2. When do we say that two angles are complementary?

IV. Translate into English:

Косинус, котангенс і косеканс називаються доповнювальними або ж ко-функціями синуса, тангенса і секанса. І навпаки, синус, тангенс і секанс називаються ко-функ­ціями косинуса, котангенса і косеканса.

TEXT 5

THE SOLUTION OF RIGHT TRIANGLES

In plane geometry it is shown that if two sides and one acute angle of a right triangle are given, the triangle can be constructed and the unknown sides and angles found by meas­urement. The same result can be obtained much more accu­rately by means of the modified definitions of the trigonomet­ric functions. Each of these expressions involves three parts of the triangle. By selecting an expression involving the two known parts and an unknown part which is to be found, an equation is obtained which can be solved for the unknown part.

Briefly, the following rules can be used to solve the un­known part of a right triangle:

To find the acute angleα, knowing the acute angle β, use the formula: α=90°—β.

To find an unknown acute angle, knowing two sides but not the other acute angle, select the proper relation involv­ing the unknown angle and the two known sides.

To find an unknown side, knowing one side and one acute angle from the relations below select the one most easily used.

Unknown side = Hypotenuse • Sine of = Hypotenuse • Cos of angle

angle opposite unknown side adjacent to unknown side

Unknown side = Known side • Tangent of = Known side-Cotangent of

angle opposite unknown side angle adjacent to unknown side

Hypotenuse = Known side • Sine of = Known side • Cosine of angle

angle opposite known side adjacent to known side

EXERCISES

I.Read the following words paying attention to the pronunciation:

formula, acute, use, compare, useful, accurate, hypote­nuse, numerical.

II.Underline all the suffixes and state to what part of speech the words belong:

measurement, relations, relate, expression, construction, opposite, adjacent, accurately.

III.Make up sentences of your own using the words and expressions given below:

definition, it is shown that, can be constructed, can be obtained, can be solved, by selecting, in solving, by means of, given triangle, unknown side.

IV.Answer the following questions:

1. What must be given in plane geometry in order to construct a right triangle? 2. What result can be obtained by means of the modified definitions of the trigonometric func­tions? 3. What rules can be used in solving the unknown part of a right triangle?

V.Translate into Ukrainian:

Trigonometric functions for an acute angle of a right tri­angle. In discussing the trigonometric functions of one of the acute angles of a right triangle, it is often advantageous to use a modification of the original definitions. If a is an acute angle of a right triangle, then

sin α = side opposite α/hypotenuse cos α = hypotenuse/side opposite α

cos α = side adjacent α/hypotenuse sec α = hypotenuse/side adjacent α

tan α = side opposite α/side adjacent α cot α = side adjacent α/side opposite α

These statements are immediately obvious if the right triangle is placed on the coordinate axes with x in standard position, applying the original definitions.

VI. Translate into English:

Розв'язання прямокутних трикутників. Нехай ABC — прямокутний трикутник, С — прямий кут, а і b — катети, протилежні гострим кутам А і В.

Тоді маємо:

Косинус гострого кута є відношенням прилеглого ка­тета до гіпотенузи:

cos A = b/c cos B = a/c

Синус гострого кута є відношенням протилежного катета до гіпотенузи:

sin A = a/c sin B = b/c

Тангенс гострого кута є відношенням протилежного катета до прилеглого:

tg A = a/b tg B = b/a

Котангенс острого кута є відношенням прилеглого катета до протилежного:

ctg A = b/c ctgA = a/b.

Сума гострых кутів у прямокутному трикутнику дорівнює 90°.

TEXT 6