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Prove the identities. \( 1 \begin{array}{l}1-\cos x \\ 1-\sin x \\ \frac{\cos ^{2} x-\cos x-\sin ^{2} x}{\cos x} \\ 2 \sin x \cdot \cos x+\sin x\end{array} \frac{1}{\tan x}-\frac{1}{\sin x} \)
Trigonometría Feb 21, 2025
Prove the identities. \( 1 \frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x} \) \( 2 \frac{\cos ^{2} x-\cos x-\sin ^{2} x}{2 \sin x \cdot \cos x+\sin x}=\frac{1}{\tan x}-\frac{1}{\sin x} \)
Trigonometría Feb 21, 2025
\begin{tabular}{l} Use the graph \( f(x)=-x^{2}+8 x+20 \) to determine the values of: \\ \hline 2.2 .1 \\ \( p \) for which \( -x^{2}+8 x+p=0 \) will have equal roots. \end{tabular}
Álgebra Feb 21, 2025
QUESTION 6 (Circulatory and respiratory system) 6.1 Describe the function of the circulatory system. (1) 6.2 Name ONE health issues involving the circulatory system. (1) 6.3 Blood flows through a network of ducts called blood vessels. Answer the questions based on the diagram below
Biología Feb 21, 2025
\( a = \frac { p ( u - \sqrt[ 3 ] { s v } ) ^ { 2 } } { t } \)
Álgebra Feb 21, 2025
1 Consider the sequence 21, \( , 36, b, 55 \) which has a second constant second difference of 1 11 Determine the values of \( a \) and \( b \) 12 Determine the \( \pi^{\text {to }} \) term of the sequence. (16) I. 3 Hence prove that the sum of any two consecutive terms in the sequence is a square number. ( \( S \) ) 2 Consider the sequence \( \frac{1}{2}+\left(\frac{1}{3}+\frac{2}{3}\right)+\left(\frac{1}{4}+\frac{2}{4}+\frac{3}{4}\right)+\left(\frac{1}{5}+\frac{2}{5}+\frac{3}{5}+\frac{4}{5}\right)+\ldots+\left(\frac{1}{200}+\frac{2}{200}+\frac{3}{200}+\ldots+\frac{199}{200}\right) \) with \( \Gamma_{2}=\frac{1}{2} \) and \( T_{2}=\left(\frac{1}{3}+\frac{2}{3}\right)=1 \) and \( T_{n}=\left(\frac{1}{200}+\frac{2}{200}+\frac{3}{200}+\ldots \frac{199}{200}\right) \) 21 Calculate the values of the third and fourth terms. 2.2 Determine the number of terms in the sequence. 2.3 What is the value of the last term in the sequence? 24 Calculate the sum of all the terms in the sequence. Prove that: \( \sum_{n=1}^{k} 2^{n-1}=2^{k}-1 \) Prove, without using a formula, that the sum of the series \( 3^{2}+3^{4}+3^{6}+\ldots \) for twenty terms
Álgebra Feb 21, 2025
\( \frac { \cos ( 180 ^ { \circ } + \theta ) \cdot \tan ( - \theta ) } { \operatorname { in } ( 360 ^ { \circ } - \theta ) \cdot \tan ( 720 ^ { \circ } + \theta ) } \)
Trigonometría Feb 21, 2025
EXERCISE 8 1. Prove that the following statements are identities. \( \begin{array}{ll}\text { (a) }(\sin \theta+\cos \theta)^{2}=1+2 \sin \theta \cdot \cos \theta & \text { (An important result to remember) } \\ \text { (b) } \frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \text { (An important result to remember) } \\ \text { (c) } \frac{1}{\sin ^{2} \theta}-\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=1 & \text { (d) } \frac{\tan x \cdot \cos x}{\sin x}=1 \\ \text { (e) } \sin ^{4} x-\cos ^{4} x=\sin ^{2} x-\cos ^{2} x & \text { (f) } \sin ^{2} \theta+\sin ^{2} \theta \cdot \tan ^{2} \theta=\tan ^{2} \theta \\ \text { (g) } \cos \theta(1+\tan \theta)=\cos \theta+\sin \theta & \text { (h) } \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}=\frac{1-\cos \theta}{1+\cos \theta} \\ \text { (i) } \frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}=\frac{2}{\sin ^{2} \theta} & \text { (j) } \frac{1}{\sin \theta}+\frac{1}{\tan \theta}=\frac{1+\cos \theta}{\sin \left(180^{\circ}-\theta\right)}\end{array} \)
Trigonometría Feb 21, 2025
QUESTION 2 2.1 Given the geometric sequence: \( 3 ; 2 ; k ; \ldots \) \( 2.1 .1 \quad \) Write down the value of the common ratio. \( 2.1 .2 \quad \) Calculate the value of \( k \) \( 2.1 .3 \quad \) Calculate the value of \( n \) if \( T_{\mathrm{*}}=\frac{128}{729} \).
Preálgebra Feb 21, 2025
\begin{tabular}{lll} 3.1 State Newton's second law in words. \\ 3.2 & Draw a free body diagram of ALL forces acting on the block. \\ 3.3 & \( \begin{array}{l}\text { Calculate the magnitude of: } \\ \text { 3.3.1 The vertical component of the applied force. } \\ \text { 3.3.2 The normal force. }\end{array} \) \\ \( \begin{array}{ll}\text { 3.3.3 Frictional force. }\end{array} \) \\ 3.4 & \( \begin{array}{l}\text { What will happen to the coefficient of frictional force between the block } \\ \text { and the surface if the angle of the applied force is decreases from } 30^{\circ} \\ \text { to } 26^{\circ} \text { ? } \\ \text { Write down only INCREASES, DECREASES or REMAINS THE SAME. }\end{array} \) \\ 3.5 & \( \begin{array}{l}\text { (2) }\end{array} \) \\ \( \begin{array}{l}\text { The block above is now removed from the horizontal surface and then } \\ \text { Placed on a slope that makes an angle of } 30^{\circ} \text { with the horizontal as } \\ \text { shown in the diagram below. }\end{array} \) \\ \hline\end{tabular}
Física Feb 21, 2025
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