Arccosine - practice problems
The arccosine (or inverse cosine) function, denoted as arccos(x) or cos⁻¹(x), returns the angle whose cosine is x. It is the inverse function of cosine, defined for input values in the domain [-1, 1] and producing output angles in the range [0, π] radians or [0°, 180°]. For example, arccos(1/2) = π/3 or 60° because cos(π/3) = 1/2. The arccosine function is not periodic like cosine but is continuous and decreasing throughout its domain. It appears in applications requiring angle determination from known ratios, including navigation, physics problems involving dot products, and solving trigonometric equations. Understanding inverse trigonometric functions is essential for complete mastery of trigonometry.Instructions: For each problem, solve carefully and show your complete working.
Number of problems found: 64
- Three angles
Find all missing angle values using the Law of Cosines, given all three sides: a = 12, b = 13, and c = 20 - ABCD rhombus
ABCD is a rhombus with sides of 10.5 cm. If the length of diagonal AC = 15.8 cm, use the cosine rule to: a. calculate the length of diagonal BD to the nearest centimetre, b. find the angles of the rhombus to the nearest degree. - Rhombus 36
Rhombus ABCD with side 8 cm long has diagonal BD 11.3 cm long. Find angle DAB. - A triangle 7
A triangle lot has the dimensions a=15 m, b=10 m, and c=20 m. What is the measure of the angle between the sides of b and c? - On a mass
Forces F₁ and F₂, each with a magnitude of 40 N, act on a mass point M. Their resultant has a magnitude of 60 N. Determine the angle between forces F₁ and F₂. - Right Triangle Angle
In a right triangle ABC, hypotenuse c = 8 cm and leg b = 4 cm. What is the size of angle α? - Circumscribed circle ABC
Triangle ABC, with sides a = 15 cm, b = 17.4 cm, and c = 21.6 cm, is circumscribed by a circle. Calculate the area of the segments determined by the sides of the triangle. - Sides ratio and angles
In triangle ABC, you know the ratio of side lengths a:b:c=3:4:6. Calculate the angle sizes of triangle ABC. - Calculate cuboid
Given cuboid ABCDEFGH. We know that |AB| = 1 cm, |BC| = 2 cm, |AE| = 3 cm. Calculate in degrees the angle size formed by the lines BG and FH . - Triangle tangent area
In the triangle ABC, b=5 cm, c=6 cm, /BAC/ = 80° are given. Calculate the sizes of the other sides and angles, and further determine the sizes of the tangent tc and the area of the triangle. - Three vertices The vertices of triangle ABC are: A[1, 2, -3], B[0, 1, 2], C[2, 1, 1]. Calculate the lengths of sides AB, AC and the angle at vertex A.
- Scalar products
The vectors a = (3, -2), b = (-1, 5) are given. Determine the vector c for which a. c = 17; b . c = 3 - Trapezoid interior angles
The area for shooting training has the shape of a trapezoid, the parallel sides of which are 36 m, 21 m long, and the remaining sides are 14 m, 16 m long. Determine the size of the interior angles with a longer base. - Parallelogram - diagonals
Suppose a parallelogram ABCD, the length of one of its diagonals is equal to that of one of its sides. What are the interior angles of this parallelogram? - Course to airport
The plane flew from airport m on a course of 132° to airport n, then from n to p on a course of 235°. The distance between the airport's mn is 380 km, np 284 km. What will be the return course to m, and what is the distance between the airport's pm? - Sin cos tan
If cos y = 0.8, 0° ≤ y ≤ 90°, find the value of (4 tan y) / (cos y-sin y) - Loonie
Loonie has three wooden sticks measuring 17 inches, 21 inches, and 25 inches. He lays them down to form a triangle. Find the measure of the angle enclosed by 17 inches and 21 inches. (Express answers to the nearest hundredths) (using the law of cosines) - A chord
In a circle radius of 6 cm, a chord is drawn 3 cm from the center. Calculate the angle subtended by the chord at the center of the circle. Hence find the length of the minor arc cut off by the chord. - Parallelogram - angle alfa
In the parallelogram ABCD the length of sides are AB = 8, BC = 5, BD = 7. Calculate the magnitude of the angle α = ∠DAB (in degrees). - Triangle side angle
The triangle ABC determines the size of the sides a and b and the magnitudes of the interior angles β and γ, given c = 1.86 m, the median to side c is 2.12 m, and the angle alpha is 40 ° 12 '.
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