Precalculus - Right Triangle Trigonometry

Right Triangles [Recap]

  • A right triangle is a special type of triangle in which the measure of one angle is exactly 90°. The measure of the other two angles sums up to 90°.
  • Each side of a right triangle has a name, namely - hypotenuse, altitude (perpendicular), and base.
  • The largest side of any right triangle is termed a hypotenuse and is always opposite to the right angle.
  • The other two sides forming the right angle are called base and altitude (perpendicular).
  • A right triangle is shown in the figure below. 

  • In the given right triangle ABC, side AC opposite to the right angle is the hypotenuse.
  • Notice that the altitude of a right triangle is defined relative to an angle. For example, relative to C side AB acts as altitude. Relative to A, side BC acts as altitude.
  • The side left after assigning hypotenuse, and altitude is designated as a base.
  • A right triangle obeys Pythagoras' theorem i.e hypotenuse2=base2+altitude2.

Trigonometric Ratios

  • In a right-angled triangle, the ratios of sides concerning any of its acute angles are known as trigonometric ratios of that particular angle.
  • In total, there are six trigonometric ratios.
  • Consider right triangle ABC (as shown in the figure below) right angled at B, the trigonometric ratios of A are defined as follows :

sine of A =altitudehypotenuse=BCAC

cosine of A =basehypotenuse=ABAC

tangent of A =altitudebase=BCAB

cotangent of A =basealtitude=ABBC

secant of A =hypotenusebase=ACAB

cosecant of A =hypotenusealtitude=ACBC

  • The trigonometric ratios defined above are abbreviated as sin A, cos A, tan A, cot A, sec A, and cosec A respectively.

Trigonometric ratios of complementary angles

  • A pair of angles whose sum is 90° called complementary angles.
  • In a right-angled triangle, the two acute angles are always complementary.
  • Thus, for a right-angled triangle, as shown in the figure:

sin θ =BCAC= cos 90°-θ 

cos θ =ABAC= sin 90°-θ

tan θ =BCAB= cot 90°-θ

cot θ =ABBC= tan 90°-θ

sec θ =ACAB= cosec 90°-θ

cosec θ =ACBC= sec 90°-θ

Solved Examples

Example 1: Find the values of sin θcos θ and tan θ for the right triangle as shown in the figure.

Solution: For the angle 'θ', hypotenuse = PQ=10 cmaltitude=QR=8 cm and base = PR = 6 cm. Thus, 

sin θ=altitudehypotenuse=810=45

cos θ=basehypotenuse=610=35

tan θ=altitudebase=86=43

Example 2: In a ABC, right angled at B, if sinA=13, then find the value of sin A·cos A·tan A.

Solution: It is given that sin A=13 but sin A=altitudehypotenuse. From here, we can conclude that altitude = k and hypotenuse = 3k, where k is any positive number. Let us now draw the triangle with the known data.

Using the Pythagorean theorem, we get, 


or,  base2=AB2=3k2-k2=2k2

so,  base=AB=k2

Now, cos A=basehypotenuse=ABAC=k2k3=23

similarly, tan A=altitudebase=BCAB=kk2=12

Therefore, sin A·cos A·tan A=13×23×12=13

Example 3: In a right triangle if sin θ+30°=cos 2θ, find the value of the acute angle θ.

Solution: It is given that sin θ+30°=cos 2θ. We can rewrite this expression as sin θ+30°=sin 90°-2θ.

Therefore, θ+30°=90°-2θ 3θ=90°-30°=60°  θ=60°3=20°

Example 4: Find the value of sin 40°cos 50°+tan 20°cot 70°-1.

Solution: sin 40°cos 50°+tan 20°cot 70°-1=sin 40°sin 90°-50°+tan 20°tan 90°-70°-1=sin 40°sin 40°+tan 20°tan 20°-1=1+1-1=1

Cheat Sheet

In any right-angled triangle, three basic trigonometric ratios defined for an acute angle 'θ' are:

  • sin θ=altitudehypotenuse
  • cos θ=basehypotenuse
  • tan θ=altitudebase

Relationship between trigonometric ratios:

  • sin θ·cosec θ=1  or  sin θ=1cosec θ  or  cosec θ=1sin θ
  • cos θ·sec θ=1  or  cos θ=1sec θ  or  sec θ=1cos θ
  • tan θ·cot θ=1  or  tan θ=1cot θ  or  cot θ=1tan θ
  • tan θ=sin θcos θ
  • cot θ=cos θsin θ

Blunder Areas

  • A triangle can never have two right angles.
  • Trigonometric ratios apply to right-angled triangles only.