The trigonometric ratio of an angle [latex]\theta[/latex] is the same for every right triangle containing the angle.
If we know one of the sides of a right triangle and any one of the other four parts, we can use trigonometry to find all the other unknown parts.
If we know one of the trigonometric ratios of an acute angle, we can find the angle using the inverse trig key on a calculator.
The exact values of trigonometric ratios of the special angles should be memorized.
Trigonometric Ratios for the Special Angles
Angle
Sine
Cosine
Tangent
[latex]30°[/latex]
[latex]\dfrac{1}{2}[/latex]
[latex]\dfrac{\sqrt{3}}{2}[/latex]
[latex]\dfrac{1}{\sqrt{3}}[/latex]
[latex]45°[/latex]
[latex]\dfrac{1}{\sqrt{2}}[/latex]
[latex]\dfrac{1}{\sqrt{2}}[/latex]
[latex]1[/latex]
[latex]60°[/latex]
[latex]\dfrac{\sqrt{3}}{2}[/latex]
[latex]\dfrac{1}{2}[/latex]
[latex]\sqrt{3}[/latex]
You can remember the trig values for the special angles if you memorize two triangles:
For the trigonometric ratios of most angles, your calculator gives approximations, not exact values.
definition
The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the third side.
The Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical notation, it can be expressed as:
[ c^2 = a^2 + b^2 ]
where [latex]ax[/latex] and [latex]b[/latex] are the lengths of the two legs of the right triangle, and [latex]c[/latex] is the length of the hypotenuse. This theorem is named after the ancient Greek mathematician Pythagoras, who was the first to prove its validity.