Formulas:

where A represents the angle of reference. 

The formulas can be remembered by:oh heck, another hour of algebra!

The formulas can be remembered by:oscar had a heap of apples

There are many such memory tricks.

Basic Trigonometry Rules: These formulas ONLY work in a right triangle. The hypotenuse is always across from the right angle. Questions usually ask for an answer to the nearest units. You will need a scientific or graphing calculator.

 

How to set up and solve a trigonometry problem when solving for a side of the triangle:

Example 1:  In right triangle ABC, hypotenuse AB=15 and angle                      A=35º.    Find leg length, BC, to the nearest tenth.

Set Up the Drawing:1. Draw a picture depicting the situation.2. Be sure to place the degrees INSIDE the

triangle.3. Place a stick figure at the angle as a point of

reference.4. Thinking of yourself as the stick figure,

label the opposite side (the side across from

you), the hypotenuse (across from the right

angle), and the adjacent side (the leftover

side).5. Notice how the values on the sides of the

triangle "pair up".  The h pairs with the 15, the o pairs with the x, but the a stands alone.  The a has no companion term.  This means that the a is NOT involved in the solution of this problem.  Cross it out!

6. This problem deals with o and h which means it is using sin A.

 

ANSWER:  8.6

Set Up the Formula:1. Place the degrees in the formula for angle A.2. Replace o and h with their companion

terms.3. Using your scientific/graphing calculator,

determine the value of the left side of the equation.  (On most scientific calculators, press 35 first and then press the sin key.  On most graphing calculators, press the sin key first and the 35 second.)

4. Solve the equation algebraically.  In this case, cross multiply and solve for x.  Or just remember that if the x is on the top, you will multiply to arrive at your answer.  If x is on the bottom, divide to arrive at your answer (see next example).

5. Round answer to the desired value.

Example 2:  In right triangle ABC, leg length BC=20 and angle       B = 41º.  Find hypotenuse length BA to the nearest hundredth.

Set up the diagram and the formula in the same manner as was done in Example 1.  You should arrive at the drawing and the formula shown here.

 

Hint:  If you are having a problem solving the equation algebraically, remember that when x is on the bottom, you must divide to arrive at your answer.  The division is always "divide BY the trig value decimal".

Hint:  Be sure your answer MAKES SENSE!!!  The hypotenuse is always the largest side in a right triangle.  So, our answer of 26.50 makes sense - it is bigger than the leg of 20.

 

You really know a lot of facts about these triangles:

In this triangle, you were given the values in black.  You also know that the angle at B

is 68º because there are 180º in every triangle.   By repositioning your reference point, all of the

following are true.  

TRIGONOMETRY OF RIGHT TRIANGLESPLANE TRIGONOMETRY is based on the fact of similar figures.  (Topic   1 .)  We saw:

Figures are similar if they are equiangularand the sides that make the equal angles are proportional.

For triangles to be similar, however, it is sufficient that they be equiangular. (Theorem 15 of "Some Theorems of Plane Geometry.") From that it follows:

Right triangles will be similar if an acute angle of one is equal to an acute angle of the other.

In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar.  Therefore the sides that make the equal angles will be proportional.  Whatever ratio CA has to AB,  FD will have to DE.  If CA were half of AB, for example, then FD would also be half of DE.

A trigonometric Table is a table of ratios of sides.  In the Table, each value of sin θ represents the ratio of the opposite side to the hypotenuse -- in every right triangle with that acute angle.

If angle θ is 28°, say, then in every right triangle with a 28° angle, itssides will be in the same ratio.  We read in the Table,

sin 28° = .469

This means that in a right triangle having an acute angle of 28°, its opposite side is 469 thousandths of the hypotenuse, which is to say, a little less than half.

It is in this sense that in a right triangle, the trigonometric ratios -- the sine, the cosine, and so on -- are "functions" of the acute angle.  They depend only on the acute angle.

Example.  Indirect measurement.   When we cannot measure things directly, we can use trigonometry.

For example, to measure the height h of a flagpole, we could measure a distance of, say, 100 feet from its base. From that point P we could then measure the angle required to sight the top .  If that angle, called theangle of elevation, turned out to be 37°, then

so that    

    h    100

 = tan 37°

so that       

so that     h = 100 × tan

37°.From the Table,

tan 37° = .754Therefore, on multiplying by 100,

h = 75.4 feet.(Lesson 4 of Arithmetic..)

All functions from one functionIf we know the value of any one trigonometric function, then -- with the aid of the Pythagorean theorem -- we can find the rest.

  Example 1.   In a right triangle, sin θ = 

  5  

13

.  Sketch the triangle, place

the ratio numbers, and evaluate the remaining functions of θ.

 

To find the unknown side x, we havex2 + 52 = 132

x2 = 169 − 25 = 144.Therefore,

x =   = 12.(Lesson 26 of Algebra.)

We can now evaluate all six functions of θ:

sin θ   = 

  5  

13

       

csc θ   = 

13 5 

cos θ   = 

1213

       

sec θ   = 

1312

tan θ   = 

  5  

12

       

cot θ   = 

12 5 

Example 2.   In a right triangle, sec θ = 4.  Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ.

To say that sec θ = 4, is to say that the hypotenuse isto the adjacent side in the ratio 4 : 1.  (4 = 

4

1)

To find the unknown side x, we havex2 +

12=42

 

x2=16 − 1 = 15.

Therefore,x =  .

We can now evaluate all six functions of θ:

sin θ   =   

4  

       

csc θ   = 

    4

cos θ   = 14

       

sec θ   = 4

tan θ   =        

 cot θ   = 

    1

  Problem 1.    In a right triangle, cos θ = 

2

5.  Sketch the triangle and

evaluate sin θ.

To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload").

Problem 2.   cot θ =  .  Sketch the triangle and evaluate csc θ.

ComplementsTwo angles are called complements of one another if together they equal a right angle.  Thus the complement of 60° is 30°.  This is the degree system of measurement in which a full circle, made up of four right angles at the center, is called 360°.  (But see Topic 12:  Radian Measure.)

Problem 3.   Name the complement of each angle.

 a)  70°   20°        b)    20°   70°        c)  45°   45°        d) θ   90° − θ

The point about complements is that, in a right triangle, the two acute angles are complementary.  For, the three angles of the right triangle are together equal to two right angles (Theorem 9); therefore, the two acute angles together will equal one right angle.

(When we come to radian measure, we will see that 90° = 

π2

, and

  therefore the complement of θ is  π2  − θ.)

CofunctionsThere are three pairs of cofunctions:

The sine and the cosineThe secant and the cosecantThe tangent and the cotangent

Here is the significance of a cofunction:A function of any angle is equal to the cofunction of its complement.

This means, for example, thatsin 80° = cos 10°.

The cofunction of the sine is the cosine.  And 10° is the complement of 80°.

Problem 4.   Answer in terms of cofunctions.a)   cos 5° = sin 85°       b)   tan 60° = cot 30°       c)   csc 12° = sec 78°

d)   sin (90° − θ) = cos θ        e)   cot θ = tan (90° − θ)

In the figure:

  sin θ = 

a

c    cos φ

= a

cThus the sine of θ is equal to the cosine of its

complement.

  sec θ = 

cb    csc φ

= cb

The secant of θ is equal to the cosecant of its complement.

  tan θ = 

ab    cot φ

= ab

The tangent of θ is equal to the cotangent of its complement.