The expression of BC in terms of p and θ is BC = p / tan θ.
By the law of the sines, we conclude that the length of QR is QR = x · (sin 2α / sin α).
By the law of the cosine we find that the triangle has the following relationship: QR² = 2 · x² · (1 + cos 2α).
How to apply trigonometric functions to describe triangles
In this question we must derive expressions for the missing lengths of triangles in terms of known sides and angles. Trigonometric functions and laws are powerful resources to derive the expressions. The first triangle is a right triangle and the length of BC can be found by definition of the tangent function:
tan θ = AB / BC
BC = AB / tan θ
BC = p / tan θ
The expression of BC in terms of p and θ is BC = p / tan θ.
The second triangle is an isosceles triangle, where PQ = PR. By using triangle properties, the measure of the angle P is equal to 180° - 2α and we obtain the following expression by the law of the sine:
PQ / sin α = QR / sin (180° - 2α)
Then,
sin (180° - 2α) = sin 180° · cos 2α - cos 180° · sin 2α
sin (180° - 2α) = sin 2α
x / sin α = QR / sin 2α
QR = x · (sin 2α / sin α)
By the law of the sines, we conclude that the length of QR is QR = x · (sin 2α / sin α).
The third case represents another isosceles triangle (PQ = QR), then the measure of the angle P is 180° - 2α. By the law of the cosine we find that:
QR² = QP² + PR² - 2 · QP · PR · cos P
QR² = 2 · x² - 2 · x² · cos (180° - 2α)
Then,
cos (180° - 2α) = cos 180° · cos 2α + sin 180° · sin 2α
cos (180° - 2α) = - cos 2α
QR² = 2 · x² · (1 + cos 2α)
By the law of the cosine we find that the triangle has the following relationship: QR² = 2 · x² · (1 + cos 2α).
To learn more on trigonometric functions: https://brainly.com/question/15706158
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