Lesson 4.2
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The document discusses trigonometric ratios in right triangles. It states that if two angles of one right triangle are congruent to two angles of another right triangle, the triangles are similar and their corresponding ratios are equal. It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) in terms of the sides of a right triangle, using both symbols and words. It also introduces the mnemonic SOHCAHTOA to remember the definitions.
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X2 t08 01 circle geometry (2012)
The document discusses circle geometry and properties of triangles. It contains three theorems about circles: 1) A circle drawn through the vertices of a right triangle with diameter the hypotenuse passes through the third vertex. 2) If two points subtend the same angle from an interval, they lie on the same circle. 3) If opposite angles of a quadrilateral are supplementary, it is cyclic. It also discusses the four centers of a triangle: the incenter, circumcenter, centroid, and orthocenter and their properties. Finally, it proves the relationship between trigonometric ratios and the sides of a triangle inscribed in a circle.
X2 T08 01 circle geometry
The document discusses circle geometry and properties related to circles. It contains two main sections: (1) Converse circle theorems which describe properties of circles related to right triangles, angles subtended at points on a chord, and cyclic quadrilaterals. (2) The four centers of a triangle - the incentre, circumcentre, centroid, and orthocentre and their properties. It also discusses the interaction between geometry and trigonometry through a property relating side lengths and sine of angles of a triangle to the diameter of its circumcircle. An example problem demonstrates properties of angles and similar triangles related to a variable point on a circle.
X2 T08 03 circle geometry (2010)
The document discusses circle geometry and properties of triangles. It contains three theorems about circles: 1) A circle drawn through the vertices of a right triangle with the hypotenuse as diameter will be tangent to the third side. 2) If two points subtend the same angle from an interval on the same side, the four points are concyclic. 3) If opposite angles of a quadrilateral are supplementary, it is cyclic. It also discusses the four centers of a triangle: the incentre, circumcentre, centroid, and orthocentre and their properties. Finally, it proves the relationship between trigonometric ratios and the sides of a triangle if the triangle is inscribed in a circle.
X2 T08 01 circle geometry
The document discusses circle geometry and properties related to circles. It contains the following key points:
(1) It describes various circle theorems and their converses, including if the diameter of a circle is the hypotenuse of a right triangle, the circle passes through the third vertex.
(2) It discusses the four centers of a triangle - the incentre, circumcentre, centroid, and orthocentre which are the points of concurrency of the angle bisectors, perpendicular bisectors, medians, and altitudes respectively.
(3) It provides an example proof showing the relationship between the sides of a triangle and sines of the angles, where the sides are equal to the diameter of
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Trigonometry is a branch of mathematics that studies triangles and the relationships between sides and angles. It deals specifically with right triangles, which have one 90 degree angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Trigonometry defines functions that express the ratios of sides, such as sine, cosine, and tangent.
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Lesson 4.2
- 1. Trig Ratios in Right Triangles
- 2. Similar Triangles If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar. If an acute angle of one right triangle is congruent to an acute angle of another right triangle, the triangles are similar, and the corresponding ratios are equal. Any two congruent angles of different right triangles will have equal ratios associated with them.
- 3. B c a C A b
- 4. B Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 5. Sine of angle A B sin A = leg opposite hypotenuse Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 6. Cosine of angle A B cos A = leg adjacent hypotenuse Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 7. Tangent of angle A B tan A = leg opposite leg adjacent Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 8. Sine of angle B B sin B = leg opposite hypotenuse Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 9. Cosine of angle B B cos B = leg adjacent hypotenuse Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 10. Tangent of angle B B tan B = leg opposite leg adjacent Leg The side Hypotenuse adjacent to angle B and opposite angle A C Leg A The side adjacent to angle A and opposite angle B
- 11. SOHCAHTO A
- 12. Trig Ratios in Right Triangles sine θ cosecant θ cosine θ secant θ tangent θ cotangent θ
- 13. Trig Ratios in Right Triangles Words Symbols Definition cosecant θ csc θ csc θ = 1 sin secant θ sec θ sec θ = 1 cos cotangent θ cot θ cot θ = 1 tan
- 14. Find all six trigonometric functions of the angle θ. 8 θ
- 15. Answers
- 16. Evaluate the remaining trigonometric functions.
- 17. Answers