Chpt 10 - Trigonometry with handwritten notes

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-1 Chapter 10 10.1 Introduction The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." 10.2 Pythagoras’ Theorem In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, i.e. 222cab= + a c b

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-2 10.3 Pythagoras’ Theorem in Three Dimensions A three-dimensional object can be described by three measurements - length, width and height. We can use Pythagoras' Theorem to find the length of the longest straw that will fit inside the box or cylinder. Example 1 A can of drink is in the shape of a cylinder with height 15 cm and radius 4 cm. What is the length of the longest straw that will fit inside the can? 10.4 Pythagorean Triples A Pythagorean triple is a set of three numbers a, b and c such that Example 2 Determine whether the numbers 6, 8 and 10 form a Pythagorean triple.

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-3 10.5 Trigonometric Ratios For the angle θ in a right-angled triangle as shown, we name the sides as: • hypotenuse (the side opposite the right angle) • adjacent (the side "next to" θ) • opposite (the side furthest from the angle) We define the three trigonometrical ratios sine θ, cosine θ, and tangent θ as follows: To remember these, many people use TOA, CAH, SOH. Example 3 Find the length of the side marked x. 35 cm x 33°

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-4 Example 4 Find the size of angle marked θ. 10.6 Angle of elevation: In surveying, the angle of elevation is the angle from the horizontal looking up to some object: 10.7 Angle of depression: The angle of depression is the angle from the horizontal looking down to some object: Example 5 The angle of elevation of an aeroplane is 23°. If the aeroplane's altitude is 2500 m, how far away is it? 80 θ° 70

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-5 10.8 Two important identities for supplementary angles are: sin sin( ) sin 180θ= °−θ = θ cos cos( ) cos 180θ= °−θ = −θ Example 6 10.9 Trigonometric Ratio of Negative Angles In general, for any negative angle A, sin (-A) = - sin A, cos (-A) = cos A, tan (-A) = - tan A Example 7 Find the values of sin (-80 o), cos (-200o) and tan (-130o). In the diagram above, BCD is a straight line, AB=8cm, AC=17cm Calculate (a) BC (b) sin∠ACD (c) cos∠ACD D A B C 17 cm 8 cm

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-6 10.10 The Sine Rule Consider the triangle ABC, D is the point such that CD is perpendicular to AB. Using the small letters a, b and c for the sides opposite angles A, B and C respectively, the Sine Rule may be derived as follows. From triangle ADC, From triangle BCD, sin A = or h = b sin A sin B = or h = a sin B Therefore, a sin B = b sin A Rearranging, = By drawing a perpendicular from A or B to the opposite side, we can show in a similar manner that = or = Combining these equations we have the Sine Rule: C c B b A a sinsinsin == In order to use the Sine Rule, the following must be known: a) two sides and an angle opposite one of them, or b) two angles and a side opposite one of them (knowing two angles and any side is sufficient because two angles, the third is easily found.) Example 8 If A = 65°, a = 20 cm, and b = 15 cm, solve the triangle. 65° A B C b = 15 a = 20 h a h b a sin A b sin B c sin C b sin B c sin C a sin A h a b c D A C B

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-7 10.11 The Cosine Rule The Cosine Rule is normally used when the Sine Rule cannot be used. There are two cases when the Sine Rule does not apply in solving triangles: a) when two sides and the included angle are known; and b) when all three sides are known. For these cases we use what is called the Cosine Rule. To derive the Cosine Rule, let’s take any general triangle ABC. From C draw CD perpendicular to AB which forms two right triangles as shown at the right. By the Pythagorean Theorem, we have: From triangle BDC, a 2 = h2 + (c − x)2 From triangle ADC, b2 = h2 + x2 Subtracting the two equations, we have a2 − b2 = c2 − 2cx. However, x = b cos A. Therefore, the above equation becomes: a2 = b2 + c2 − 2bc cos A. By drawing a perpendicular from A or B to the opposite, we can derive the other two forms of the Cosine Rule in a similar manner. In summary, the Cosine Rule can take any of the following forms: a 2 = b2 + c2 − 2bc cos A b2 = a2 + c2 − 2ac cos B c2 = a2 + b2 − 2ab cos C Example 9 If a = 37.5, b = 28.2, and c = 11.4, solve the triangle. C c x c − x h a b D A B b = 28.2 a = 37.5 c = 11.4 A C B

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-8 10.12 Bearing The true bearing to a point is the angle measured in degrees in a clockwise direction from the north line. We will refer to the true bearing simply as the bearing. For example, the bearing of point P is 065º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass at O with the point P (i.e. OP). The bearing of point Q is 300º which is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass at O with the point Q (i.e. OQ). Note: The bearing of a point is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass with the point. A bearing is used to represent the direction of one point relative to another point. For example, the bearing of A from B is 065º. The bearing of B from A is 245º. Note: • Three figures are used to give bearings. • All bearings are measured in a horizontal plane. Example 10 State the bearing of the point P in each of the following diagrams:

Pythagoras’s Theorem and Trigonometry Mathematics Preparatory Course for Direct Entry to Higher Nitec 10-9 10.13 Three-Dimensional Problems To solve a three-di