# What are sine, cosine, and tangent?

To better understand certain problems involving aircraft and propulsion it is necessary to use some mathematical ideas from trigonometry, the study of triangles. Let us begin with some definitions and terminology which we will use on this slide.

A right triangle is a three sided figure with one angle equal to 90 degrees. A 90 degree angle is called a right angle which gives the right triangle its name. We pick one of the two remaining angles and label it c and the third angle we label d.

The sum of the angles of any triangle is equal to 180 degrees. If we know the value of c, we then know that the value of d:

• 90 + c + d = 180
• d = 180 – 90 – c
• d = 90 – c

We define the side of the triangle opposite from the right angle to be the hypotenuse. It is the longest side of the three sides of the right triangle. The word “hypotenuse” comes from two Greek words meaning “to stretch”, since this is the longest side.

We label the hypotenuse with the symbol h. There is a side opposite the angle c which we label o for “opposite”. The remaining side we label a for “adjacent”.

The angle c is formed by the intersection of the hypotenuse h and the adjacent side a.

We are interested in the relations between the sides and the angles of the right triangle. Let us start with some definitions. We will call the ratio of the opposite side of a right triangle to the hypotenuse the sine and give it the symbol sin.

sin = o / h

The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos.

cos = a / h

Finally, the ratio of the opposite side to the adjacent side is called the tangent and given the symbol tan.

tan = o / a

We claim that the value of each ratio depends only on the value of the angle c formed by the adjacent and the hypotenuse. To demonstrate this fact, let's study the three figures in the middle of the page. In this example, we have an 8 foot ladder that we are going to lean against a wall.

The wall is 8 feet high, and we have drawn white lines on the wall and blue lines along the ground at one foot intervals. The length of the ladder is fixed. If we incline the ladder so that its base is 2 feet from the wall, the ladder forms an angle of nearly 75.5 degrees degrees with the ground.

The ladder, ground, and wall form a right triangle. The ratio of the distance from the wall (a – adjacent), to the length of the ladder (h – hypotenuse), is 2/8 = .25. This is defined to be the cosine of c = 75.5 degrees.

(On another page we will show that if the ladder was twice as long (16 feet), and inclined at the same angle(75.5 degrees), that it would sit twice as far (4 feet) from the wall. The ratio stays the same for any right triangle with a 75.5 degree angle.

) If we measure the spot on the wall where the ladder touches (o – opposite), the distance is 7.745 feet. You can check this distance by using the Pythagorean Theorem that relates the sides of a right triangle:

1. h^2 = a^2 + o^2
2. o^2 = h^2 – a^2
3. o^2 = 8^2 – 2^2
4. o^2 = 64 – 4 = 60
5. o = 7.745

The ratio of the opposite to the hypotenuse is .967 and defined to be the sine of the angle c = 75.5 degrees.

Now suppose we incline the 8 foot ladder so that its base is 4 feet from the wall. As shown on the figure, the ladder is now inclined at a lower angle than in the first example. The angle is 60 degrees, and the ratio of the adjacent to the hypotenuse is now 4/8 = .5 .

Decreasing the angle c increases the cosine of the angle because the hypotenuse is fixed and the adjacent increases as the angle decreases. If we incline the 8 foot ladder so that its base is 6 feet from the wall, the angle decreases to about 41.4 degrees and the ratio increases to 6/8, which is .75.

As you can see, for every angle, there is a unique point on the ground that the 8 foot ladder touches, and it is the same point every time we set the ladder to that angle. Mathematicians call this situation a function.

The ratio of the adjacent side to the hypotenuse is a function of the angle c, so we can write the symbol as cos(c) = value.

Notice also that as the cos(c) increases, the sin(c) decreases. If we incline the ladder so that the base is 6.938 feet from the wall, the angle c becomes 30 degrees and the ratio of the adjacent to the hypotenuse is .866. Comparing this result with example two we find that:

cos(c = 60 degrees) = sin (c = 30 degrees)

sin(c = 60 degrees) = cos (c = 30 degrees)

We can generalize this relationship:

sin(c) = cos (90 – c)

90 – c is the magnitude of angle d. That is why we call the ratio of the adjacent and the hypotenuse the “co-sine” of the angle.

sin(c) = cos (d)

Since the sine, cosine, and tangent are all functions of the angle c, we can determine (measure) the ratios once and produce tables of the values of the sine, cosine, and tangent for various values of c. Later, if we know the value of an angle in a right triangle, the tables will tell us the ratio of the sides of the triangle.

If we know the length of any one side, we can solve for the length of the other sides. Or if we know the ratio of any two sides of a right triangle, we can find the value of the angle between the sides. We can use the tables to solve problems.

Some examples of problems involving triangles and angles include the forces on an aircraft in flight, the application of torques, and the resolution of the components of a vector.

Here are tables of the sine, cosine, and tangent which you can use to solve problems.   Activities: Guided Tours     Beginner's Guide Home Page

## Math Scene – Trigonometry sine, cosine and tangent – Lesson 1

 © 2008  Rasmus ehf and Jóhann Ísak Print

ABC is a right angled triangle The angle  A  is  30 degrees. We write this as:
• a    is the symbol for the side opposite angle A
• b    is the symbol for the side opposite angle B
• c    is the symbol for the side opposite angle C
• Similar triangles are triangles in which all the angles in one triangle are equal to the angles in the other triangle These two triangles are similar. The ratio between two sides in one triangle is equal to the ratio between the corresponding sides in the other triangle.
1. Using the notation in the above triangles we get the following:
2. 3. The ratio depends on the size of the angle.
4. Tangent The ratio called tangent (tan) of an acute angle in a right angled triangle is defined as the ration between the side opposite the angle and the side adjacent to the angle .

### Example1Find the angle  A First  Tan A = 3/4 = 0.75

We need to use the inverse function for tan,  tan-1, to find the angle. This function is on the same key on the calculator as the tan function (shift tan).

• We use the following sequence of commands:
• shift    –     tan-1   0.75     = 37º
• Try the following on your calculator to see the difference between tan and  tan-1:
• angle    →    ratio                            ratio  →     angle

tan 37º     =     0.75                   tan-1 0.75    =  37º

### Example 2     Find the side b tan 37º = 4/b tan 37º · b = 4 0.75· b = 4 b=5.3

Síne The sine (sin) of an acute angle  in a right angled triangle is the ratio between the side opposite  the angle and the hypotenuse of the triangle.

### Example3     Find the angle A giving your answer to the nearest degree. sin A = 3/5 = 0.6 gives

## What is Sine, Cosine and Tangent?

Trigonometry deals with the sides and angles in triangles and the relationship between them. In a right-angled triangle, the sides are named according to the each of the acute angles,

The longest side in the triangle is called the hypotenuse – it is opposite the angle of #90°#. The side next to an angle (one of its arms) is called the adjacent side while the side on the other side from the angle is called the opposite side.

In trigonometry the lengths of the 3 sides are compared in the form of ratios. Think of them as fractions.

• If the sides are of length # 3, 4 and 5#, then we can write #6# ratios as a way of comparing them:
• #3/5,” ” 4/5, ” ” 4/3,” ” 5/3,” ” 4/5,” ” 3/5#
• When the ratio involves the sides: #”opposite”/”hypotenuse”# it is called Sine .
• When the ratio involves the sides: #”adjacent”/”hypotenuse”# it is called Cosine.
• When the ratio involves the sides: #”opposite”/”adjacent”# it is called Tangent.
• The ratios are always found with reference to an angle and they represent a value.

#sin 30° = 0.5#
This means that if a triangle has an angle of #30°#, then the side opposite the angle will be #50%# of the length of the hypotenuse. This only applies for a #30°# angle.

#cos 45° = 0.707#

This means that for an angle of #45°#, the length of the adjacent side will be #70.7%# of the length of the hypotenuse.

#tan 60° = 1.73#

This means that for an angle of #60°#, the side opposite that angle will be #173%# of the length of the side adjacent to the angle,

## Sine, Cosine and Tangent

This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle.

The Sine, Cosine and Tangent functions express the ratios of sides of a right triangle.

Answer: sine of an angle is always the ratio of the \$\$frac{opposite side}{hypotenuse} \$\$.

\$
sine(angle) = frac{ ext{opposite side}}{ ext{hypotenuse}}
\$

\$\$
sin(angle
ed L) = frac{opposite }{hypotenuse}
\
sin(angle
ed L) = frac{9}{15}
\$\$

\$\$
sin(angle
ed K) = frac{opposite }{hypotenuse}
\
sin(angle
ed K)= frac{12}{15}
\$\$

Remember: When we use the words 'opposite' and 'adjacent,' we always have to have a specific angle in mind.

For those comfortable in “Math Speak”, the domain and range of Sine is as follows.

• Domain of Sine = all real numbers
• Range of Sine = {-1 ≤ y ≤ 1}

The sine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key sine values that span the entire range of values.

 Angle Sine of the Angle 270° sin (270°) = -1 (smallest value that sine can have) 330° sin (330°) = -½ 0° sin(0°) = 0 30° sin(30°) = ½ 90° sin(90°) = 1 (greatest value that sine can have)

The cosine of an angle is always the ratio of the (adjacent side/ hypotenuse).

\$
cosine(angle) = frac{ ext{adjacent side}}{ ext{hypotenuse}}
\$

\$\$
cos(angle
\
cos(angle
ed L) = frac{12}{15}
\$\$

\$\$
cos(angle
\
cos(angle
ed K) = frac{9}{15}
\$\$

For those comfortable in “Math Speak”, the domain and range of cosine is as follows.

• Domain of Cosine = all real numbers
• Range of Cosine = {-1 ≤ y ≤ 1}

The cosine of an angle has a range of values from -1 to 1 inclusive. Below is a table of values illustrating some key cosine values that span the entire range of values.

 Angle Cosine of the Angle 0° cos (0°) = 1 (greatest value that cosine can ever have) 60° cos (60°) =½ 90° cos(90°) = 0 120° cos(120°) = -½ 180° cos(180°) = -1 (smallest value that cosine can ever have)

The tangent of an angle is always the ratio of the (opposite side/ adjacent side).

\$
tangent(angle) = frac{ ext{opposite side}}{ ext{adjacent side}}
\$

\$\$
tan(angle
ed L) = frac{opposite }{adjacent }
\
tan(angle
ed L) = frac{9}{12}
\$\$

\$\$
tan(angle
ed K) = frac{opposite }{adjacent }
\
tan(angle
ed K) = frac{12}{9}
\$\$

In the triangles below, identify the hypotenuse and the sides that are opposite and adjacent to the shaded angle.

• Hypotenuse = AB Opposite side = BC
1. Hypotenuse = AC Opposite side = BC
• Hypotenuse = YX Opposite Side = ZX
1. Hypotenuse = I Side opposite of A = H
2. Side adjacent to A = J

Identify the hypotenuse, and the opposite and adjacent sides of \$\$ angle ACB \$\$.

First, remember that the middle letter of the angle name (\$\$ angle A
ed C B \$\$) is the location of the angle.

Second: The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle — which in this case is the red angle in the picture.

Identify the hypotenuse, and the opposite and adjacent sides of \$\$ angle RPQ \$\$.

First, remember that the middle letter of the angle name (\$\$ angle R
ed P Q \$\$) is the location of the angle.

Second: The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle — which in this case is the red angle in the picture.

Identify the hypotenuse, and the opposite and adjacent sides of \$\$ angle BAC \$\$.

First, remember that the middle letter of the angle name (\$\$ angle B
ed A C \$\$) is the location of the angle.

Second: The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle — which in this case is the red angle in the picture.

Identify the side that is opposite of \$\$angle\$\$IHU and the side that is adjacent to \$\$angle\$\$IHU.

First, remember that the middle letter of the angle name (\$\$ angle I
ed H U \$\$) is the location of the angle.

Second: The key to solving this kind of problem is to remember that 'opposite' and 'adjacent' are relative to an angle of the triangle — which in this case is the red angle in the picture.

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## SINE, COSINE & TANGENT

°

°

°

SINE, COSINE & TANGENT

Here is a mnemonic phrase from Giles Marlow of Woking, Surrey for the trigonometrical ratios sine (sin), cosine (cos) and tangent (tan) of any unknown angle Ø within a right-angled triangle:

SOH – CAH – TOA!

Pronounced “…soaker toe-er…” where:
Sin Ø = Opposite/Hypotenuse
Tan Ø = Opposite/Adjacent sides Alternatives

Trigonometry is the branch of mathematics dealing with the measurement of sides and angles of triangles. A right-angled triangle has three sides (the two at 90° to each other usually shown as horizontal and vertical sides, with the remaining third side being the hypotenuse).

The sine of an angle is the ratio between the side opposite the angle concerned and the hypotenuse, while the cosine of the same angle is the ratio between the other remaining side (ie.

the one adjacent to the angle) and the hypotenuse, and the tangent of the same angle is the ratio between the opposite and adjacent sides.

Natural tables are used to convert sine, cosine and tangent values into actual degrees and vice-versa. The ratio formulas can be transposed (into Opp=Hyp*Sin, A=H*C and O=A*T) so that one can always find (1) an angle given any two sides and (2) a side given an angle and one other side. Otherwise Pythagoras' Theorem is used to find a side given any other two sides.

One alternative mnemonic for the ratios is:

Oh Heck – Another Hour Of Algebra! “
Or O/H (= Sin), A/H (= Cos), O/A (= Tan)

Mark Alcock has never forgotten the variation his old maths teacher gave him for remembering the rules of tan, cos and sine this way…

To Oil A Car Always Have Some Oil Handy

For a non-right-angled triangle, different ratio formulas apply, leading to another established mnemonic:

plusAll Stations To Coventry… “ (ie. All+, Sin+, Tan+, Cos+)

• To understand its significance, consider the different trigonometrical ratios aplying to non-right-angled triangles:
• The sine rule for: (1) a side when one side and two angles are known, or (2) an angle knowing one angle and two sides:
• side a/sin A (ie. angle A opposite side a) = side b/sin B = side c/sin C
• The cosine rule for:
(1) a third side when two sides and the included angle A are known, or (2) an angle A when all three sides are known:
• side a²=b²+c²-(2bc * cos A)
from which cos A=(b²+c²-a²)/2bc
• Also in a non-right-angled triangle one angle may be obtuse (ie. greater than 90%), whereupon one must deduct it from 180° and make its cosine value negative:
• sin Ø=sin(180°-Ø) and
cos Ø= -[minus]cos(180°-Ø)

The cosine of a obtuse angle is negative because the angle lies in the second quadrant of an imaginary circle.

A quadrant is a quarter of a circle, and measuring angles in an anti-clockwise direction between a radial startpoint X (equivalent to 3 on a clock face) and another radial point P on the circle, values are positive for all the functions of an angle Ø in the 1st quadrant (P lying between 3-12, or Ø up to 90°) but only the sines in the 2nd quadrant (12-9, or 90-180°), only the cosines in the 3rd quadrant (9-6) and only the tangents in the 4th quadrant (6-3). All other functions in each quadrant are negative. This is summarised in the mnemonic sentence:

plusAll Stations To Coventry… “ (ie. All+, Sin+, Tan+, Cos+)

Having now come so far, it seems appropriate to end with the formula that summarises the overall relationship between sine, cosine and tangent values. Easily transposeable, it is best remembered “mnemonically” by recalling its parts in straight S,C,T order:

– SOCKET –
sin over cosine equals tangent!
ie.( sin Ø / cos Ø ) = tan Ø