You will need
- - knowledge of the Pythagorean theorem, theorem of cosines;
- - trigonometric identities;
- calculator or tables Bradis.
With cosine, you can find any of the sides of a right triangle. To do this, use the mathematical relationship that States that the cosine of an acute angle triangle is the ratio of adjacent sides to the hypotenuse. Therefore, knowing the acute angle of a right triangle, find the side of it.
For example, the hypotenuse of a right triangle is 5 cm and the acute angle 60º with her. Locate adjacent to the acute corner side. To do this, use the definition of cosine cos(α)= b/a, where a is the hypotenuse of a right triangle, b is the side adjacent to the angle α. Then its length will be equal to b=a∙cos(α). Substitute values b=5∙cos(60º)= 5∙0,5=2,5 cm
The third side C, which is the second leg find using the Pythagorean theorem c=√(52-2,52)≈4.33 cm
Using the spherical law of cosines, you can find sides of triangles if you know two sides and the angle between them. To find the third side, find the sum of the squares of the two known sides, subtract from it twice the product multiplied by the cosine of the angle between them. From the result, extract the square root.
Example In triangle two sides are equal a=12 cm, b=9 cm Angle between them is 45º. Find the third side c. To find the third side use the cosine theorem c=√(a2+b2-a∙b∙cos(α)). Performing a lookup will get c=√(122+92-12∙9∙cos(45º))≈12,2 cm
When solving problems with cosines, use of identity, allowing to pass away from this trigonometric function to another, and Vice versa. The basic trigonometric identity: cos2(α)+sin2(α)=1; the ratio of tangent and cotangent: tg(α)=sin(α)/cos(α), ctg(α)=cos(α)/sin(α), etc. To find the value of the cosines of the angles use a special calculator or table Bradis.
Advice 2 : How to find the tangent of an angle in a triangle
The tangent of an angle, like other trigonometric functions, expresses the relationship between the sides and angles of a right triangle. The use of trigonometric functions allows us to replace quantities in the degree measurements on the linear parameters.
In the presence of protractor angle triangle can be measured and the table Bradis to find the tangent ratio. If it is not possible to determine the degree measure of an angle, determine its tangent using the measurements of the linear gradient shape. To do this, the support construction from an arbitrary point on one of the sides of the angle, drop a perpendicular on the other side. Measure the distance between the ends of the perpendiculars on the sides of the angle, record the measurement in the numerator of the fraction. Now measure the distance from the vertex of the given angle to the vertex angle, i.e. to the point on the side of the angle, which was lowered perpendicular. The number you write in the denominator. Based on the results of measurements of the fraction is equal to the tangent of an angle.
The tangent of the angle can be determined by calculation as the ratio of he opposite over adjacent. You can also calculate the tangent using the direct trigonometric functions of the angle sine and cosine. The tangent of an angle is the ratio of the sine of that angle to its cosine. Unlike continuous functions of sine and cosine, the tangent is discontinuous and not defined at the angle of 90 degrees. If there is a zero angle of its tangent equals zero. From ratios of a right triangle it is obvious that the 45 degree angle has a tangent equal to one, since the legs of this right triangle are equal.
For small values of angle from 0 to 90 degrees its tangent is positive, since the sine and cosine in that interval is positive. Limits of change of the tangent in this area - from zero to infinitely high values at angles close to straight. Negative values of the angle of the tangent changes sign. The graph of the function Y=tg(x) in the interval -90°