Trigonometry studies the relationships between sides and angles of triangles. The sine, cosine, and tangent functions describe ratios of sides in right triangles. Spherical trigonometry applies these concepts to triangles on the surface of a sphere. It is used to solve problems involving latitude, longitude, and great circles. Key formulas in spherical trigonometry include the cosine rule, sine rule, and cotangent law, which relate angles and sides of spherical triangles.

1. TRIGONOMETRY
Applications of Plane and SphericalTrigonometry

2. INDEX

3. WHAT DOES
ITSTUDY?
 Trigonometry studies the
relationships between side
lengths and angles of
triangles .
 In a rectangle triangle the
ratios are given by the sine,
cosine and tangent.
To solve a triangle we need to
know three elements, one of
them have to be a side.

4. SINE
 Sine function (sin) is the ratio of the side opposite the angle to the
hypotenuse:
sin 𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
𝑎
𝑐

5. COSINE
 Cosine function (cos) defined as the ratio of the adjacent leg ( the
side of the triangle joining the angle to the right angle) to the
hypotenuse:
cos 𝐴 =
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
𝑏
𝑐

6. TANGENT
 Tangent function (tan), defined as the ratio of the opposite leg to
the adjacent leg:
tan 𝐴 =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
=
𝑎
𝑏
=
𝑎/𝑐
𝑏/𝑐
=
Sin 𝐴
𝐶𝑜𝑠 𝐴

7. HYPOTENUSE
The hypotenuse is the side opposite to the 90 degree
angle in a right triangle; it is the longest side of the
triangle and one of the two sides adjacent to angleA.The
adjacent leg is the other side that is adjacent to angleA.
The opposite side is the side that is opposite to angle A.

8. RESOLUTION
b= 𝑎 sin 𝛽 ; b= 𝑎 cos 𝛾
c= 𝑎 sin 𝛾 ; c= 𝑎 cos 𝛽
b= 𝑐 tan 𝛽
c= 𝑏 tan 𝛾
Ex:
𝑏 = 15 3 𝛽 = 60° 𝛾 = 30°
𝑐 = b ∙ tan 𝛾 = 15 3 ∙ tan 30° = 15 3 ∙
1
3
= 15
𝑎 = 𝑏2 + 𝑐2= 675 + 225= 900 = 30

9. SPHERICAL
TRIGONOMETRY
To talk about spherical trigonometry we need to introduce the concept the
sphere of the Earth:
 The axis: is an imaginary line passing through the centre of the Earth and
connects the two poles;
 The Equator: represents the maximum circumference of the Earth and it’s the
Prime Parallel;
 Greenwich: is the arc of maximum circumference running from the North Pole
to the South Pole and it’s the Prime Meridian;
 Parallels: are smaller circle than the Equator and they gradually diminish in
lenght as they get nearer to the poles, they’re lines running from east to west;
 Meridians: are maximum circumference or great circle passing through the
poles.

11. GEOGRAPHICAL
COORDINATES
LATITUDE
 Latitude is the distance 0f a
place from the Equator.
 The latitude of a place is the
arc of the meridian found
between the parallel passing
through that place and the
Equator.
LONGITUDE
 Longitude is the distance of a
place from the Greenwich
meridian.
 The longitude of a place is the arc
of the Equator found between
the Greenwich meridian and the
meridian passing through that
place.

12. The Spherical trigonometry is the science that studies the relationships
between the sides and angles of the spherical triangles.
 GREAT CIRCLE: it corresponds to an imaginary plane passing through
the centre of the Earth which divides the terrestrial globe into two
emispheres.
 SMALL CIRCLE: it is an other circle formed on the surface of the sphere
which does not pass through the centre of the Earth.
 SPHERICALTRIANGLE: it is the triangle formed by three arcs of great
circles on a sphere.
To solve a spherical triangle we need to know some of the elements of which
it’s made up, due to that he have got three sides and three angles .

13. COSINEAND
SINE RULES
 To solve the triangle we usually use the cosine rules, sine rules and
the cotangent rules.
 COSINE RULE: the cosine rule is given by the formulas :
cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝛼
cos 𝑏 = cos 𝑎 cos 𝑐 + sin 𝑎 sin 𝑐 cos 𝛽
cos 𝑐 = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏 cos 𝛾
SINE RULE: that is given by the formula:
sin 𝛼
sin 𝑎
=
sin 𝛽
sin 𝑏
=
sin 𝛾
sin 𝑐

14. LAWOF
COTANGENT
 The law is given by a formula that deals the relaionship between
four elements of a spherical triangle ( two sides and two angles for
example)
cot(. . ) sin(. . ) = cos(. . ) cos(. . ) + sin(. . ) cot(. . )
To insert the elements inside the formula
we have to:
• Start from a side
• Go to one of the other side
• Go in the angle in common
• Go in the other angle.